3.
Lessons In Electric Circuits, Volume V – Reference
By Tony R. Kuphaldt
Fourth Edition, last update April 19, 2007
4.
i
c 2000-2009, Tony R. Kuphaldt
This book is published under the terms and conditions of the Design Science License. These
terms and conditions allow for free copying, distribution, and/or modiﬁcation of this document
by the general public. The full Design Science License text is included in the last chapter.
As an open and collaboratively developed text, this book is distributed in the hope that
it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the Design Science
License for more details.
Available in its entirety as part of the Open Book Project collection at:
www.ibiblio.org/obp/electricCircuits
PRINTING HISTORY
• First Edition: Printed in June of 2000. Plain-ASCII illustrations for universal computer
readability.
• Second Edition: Printed in September of 2000. Illustrations reworked in standard graphic
(eps and jpeg) format. Source ﬁles translated to Texinfo format for easy online and printed
publication.
• Third Edition: Equations and tables reworked as graphic images rather than plain-ASCII
text.
• Fourth Edition: Printed in XXX 2001. Source ﬁles translated to SubML format. SubML is
a simple markup language designed to easily convert to other markups like LTEX, HTML,
A
or DocBook using nothing but search-and-replace substitutions.
10.
2 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS
1.11 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.12 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.1 DC circuit equations and laws
1.1.1 Ohm’s and Joule’s Laws
Ohm’s Law
E = IR I= E R= E
R I
Joule’s Law
2
P = IE P= E P = I2R
R
Where,
E = Voltage in volts
I = Current in amperes (amps)
R = Resistance in ohms
P = Power in watts
NOTE: the symbol ”V” (”U” in Europe) is sometimes used to represent voltage instead of
”E”. In some cases, an author or circuit designer may choose to exclusively use ”V” for voltage,
never using the symbol ”E.” Other times the two symbols are used interchangeably, or ”E” is
used to represent voltage from a power source while ”V” is used to represent voltage across a
load (voltage ”drop”).
1.1.2 Kirchhoff’s Laws
”The algebraic sum of all voltages in a loop must equal zero.”
Kirchhoff’s Voltage Law (KVL)
”The algebraic sum of all currents entering and exiting a node must equal zero.”
Kirchhoff’s Current Law (KCL)
11.
1.2. SERIES CIRCUIT RULES 3
1.2 Series circuit rules
• Components in a series circuit share the same current. Itotal = I1 = I2 = . . . In
• Total resistance in a series circuit is equal to the sum of the individual resistances, mak-
ing it greater than any of the individual resistances. Rtotal = R1 + R2 + . . . Rn
• Total voltage in a series circuit is equal to the sum of the individual voltage drops. Etotal
= E1 + E2 + . . . En
1.3 Parallel circuit rules
• Components in a parallel circuit share the same voltage. Etotal = E1 = E2 = . . . En
• Total resistance in a parallel circuit is less than any of the individual resistances. Rtotal
= 1 / (1/R1 + 1/R2 + . . . 1/Rn )
• Total current in a parallel circuit is equal to the sum of the individual branch currents.
Itotal = I1 + I2 + . . . In
1.4 Series and parallel component equivalent values
1.4.1 Series and parallel resistances
Resistances
Rseries = R1 + R2 + . . . Rn
1
Rparallel =
1 1 1
R1 + R2 + . . . Rn
12.
4 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS
1.4.2 Series and parallel inductances
Inductances
Lseries = L1 + L2 + . . . Ln
1
Lparallel =
1 1 1
L1 + L2 + . . . Ln
Where,
L = Inductance in henrys
1.4.3 Series and Parallel Capacitances
Capacitances
1
Cseries =
1 1 1
C1 + C2 + . . . Cn
Cparallel = C1 + C2 + . . . Cn
Where,
C = Capacitance in farads
1.5 Capacitor sizing equation
εA
C=
d
Where,
C = Capacitance in Farads
ε = Permittivity of dielectric (absolute, not
relative)
A = Area of plate overlap in square meters
d = Distance between plates in meters
13.
1.5. CAPACITOR SIZING EQUATION 5
ε = ε0 K
Where,
ε0 = Permittivity of free space
ε0 = 8.8562 x 10-12 F/m
K= Dielectric constant of material
between plates (see table)
Dielectric constants
Dielectric K Dielectric K
Vacuum 1.0000 Quartz, fused 3.8
Air 1.0006 Wood, maple 4.4
PTFE, Teflon 2.0 Glass 4.9-7.5
Mineral oil 2.0 Castor oil 5.0
Polypropylene 2.20-2.28 Wood, birch 5.2
ABS resin 2.4 - 3.2 Mica, muscovite 5.0-8.7
Polystyrene 2.45-4.0 Glass-bonded mica 6.3-9.3
Waxed paper 2.5 Poreclain, steatite 6.5
Transformer oil 2.5-4 Alumina Al2O3 8-10.0
Wood, oak 3.3 Water, distilled 80
Hard Rubber 2.5-4.8 Ta2O5 27.6
Silicones 3.4-4.3 Ba2TiO3 1200-1500
Bakelite 3.5-6.0 BaSrTiO3 7500
A formula for capacitance in picofarads using practical dimensions:
0.0885K(n-1) A 0.225K(n-1)A’
C= =
d d’
A
Where, d
C = Capacitance in picofarads
K = Dielectric constant
A = Area of one plate in square centimeters
A’ = Area of one plate in square inches
d= Thickness in centimeters
d’ = Thickness in inches
n= Number of plates
14.
6 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS
1.6 Inductor sizing equation
N2µA
L=
l
µ = µr µ0
r
Where, l
L = Inductance of coil in Henrys
N = Number of turns in wire coil (straight wire = 1)
µ = Permeability of core material (absolute, not relative)
µr = Relative permeability, dimensionless ( µ0=1 for air)
-6
µ0 = 1.26 x 10 T-m/At permeability of free space
A = Area of coil in square meters = πr2
l = Average length of coil in meters
Wheeler’s formulas for inductance of air core coils which follow are usefull for radio fre-
quency inductors. The following formula for the inductance of a single layer air core solenoid
coil is accurate to approximately 1% for 2r/l < 3. The thick coil formula is 1% accurate when
the denominator terms are approximately equal. Wheeler’s spiral formula is 1% accurate for
c>0.2r. While this is a ”round wire” formula, it may still be applicable to printed circuit spiral
inductors at reduced accuracy.
r c c
r r
l
2 2
Nr
L=
9r + 10⋅l l
0.8N2r2 N2r2
L= L=
6r + 9⋅l + 10c 8r + 11c
Where,
L = Inductance of coil in microhenrys
N = Number of turns of wire
r = Mean radius of coil in inches
l = Length of coil in inches
c = Thickness of coil in inches
15.
1.7. TIME CONSTANT EQUATIONS 7
The inductance in henries of a square printed circuit inductor is given by two formulas
where p=q, and p=q.
L = 85⋅10-10DN5/3 L = 27⋅10-10(D8/3/p5/3)(1+R-1)5/3
p
Where, Where,
D
D = dimension, cm D = coil dimension in cm
N = number turns q N = number of turns
p=q R= p/q
The wire table provides ”turns per inch” for enamel magnet wire for use with the inductance
formulas for coils.
AWG turns/ AWG turns/ AWG turns/ AWG turns/
gauge inch gauge inch gauge inch gauge inch
10 9.6 20 29.4 30 90.5 40 282
11 10.7 21 33.1 31 101 41 327
12 12.0 22 37.0 32 113 42 378
13 13.5 23 41.3 33 127 43 421
14 15.0 24 46.3 34 143 44 471
15 16.8 25 51.7 35 158 45 523
16 18.9 26 58.0 36 175 46 581
17 21.2 27 64.9 37 198
18 23.6 28 72.7 38 224
19 26.4 29 81.6 39 248
1.7 Time constant equations
1.7.1 Value of time constant in series RC and RL circuits
Time constant in seconds = RC
Time constant in seconds = L/R
16.
8 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS
1.7.2 Calculating voltage or current at speciﬁed time
Universal Time Constant Formula
1
Change = (Final-Start) 1 -
et/τ
Where,
Final = Value of calculated variable after infinite time
(its ultimate value)
Start = Initial value of calculated variable
e = Euler’s number ( 2.7182818)
t = Time in seconds
τ = Time constant for circuit in seconds
1.7.3 Calculating time at speciﬁed voltage or current
1
t=τ ln
Change
1-
Final - Start
1.8 AC circuit equations
1.8.1 Inductive reactance
XL = 2πfL
Where,
XL = Inductive reactance in ohms
f = Frequency in hertz
L = Inductance in henrys
17.
1.8. AC CIRCUIT EQUATIONS 9
1.8.2 Capacitive reactance
XC = 1
2πfC
Where,
XC = Inductive reactance in ohms
f = Frequency in hertz
C = Capacitance in farads
1.8.3 Impedance in relation to R and X
ZL = R + jXL
ZC = R - jXC
1.8.4 Ohm’s Law for AC
E = IZ I= E Z= E
Z I
Where,
E = Voltage in volts
I = Current in amperes (amps)
Z = Impedance in ohms
1.8.5 Series and Parallel Impedances
Zseries = Z1 + Z2 + . . . Zn
1
Zparallel =
1 1 1
Z1 + Z2 + . . . Zn
NOTE: All impedances must be calculated in complex number form for these equations to
work.
18.
10 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS
1.8.6 Resonance
1
fresonant =
2π LC
NOTE: This equation applies to a non-resistive LC circuit. In circuits containing resistance
as well as inductance and capacitance, this equation applies only to series conﬁgurations and
to parallel conﬁgurations where R is very small.
1.8.7 AC power
E2
P = true power P = I2R P=
R
Measured in units of Watts
E2
Q = reactive power Q = I2X Q=
X
Measured in units of Volt-Amps-Reactive (VAR)
E2
S = apparent power S = I2Z S= S = IE
Z
Measured in units of Volt-Amps
P = (IE)(power factor)
S= P2 + Q2
Power factor = cos (Z phase angle)
20.
12 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS
1.10 Metric preﬁxes and unit conversions
• Metric preﬁxes
• Yotta = 1024 Symbol: Y
• Zetta = 1021 Symbol: Z
• Exa = 1018 Symbol: E
• Peta = 1015 Symbol: P
• Tera = 1012 Symbol: T
• Giga = 109 Symbol: G
• Mega = 106 Symbol: M
• Kilo = 103 Symbol: k
• Hecto = 102 Symbol: h
• Deca = 101 Symbol: da
• Deci = 10−1 Symbol: d
• Centi = 10−2 Symbol: c
• Milli = 10−3 Symbol: m
• Micro = 10−6 Symbol: µ
• Nano = 10−9 Symbol: n
• Pico = 10−12 Symbol: p
• Femto = 10−15 Symbol: f
• Atto = 10−18 Symbol: a
• Zepto = 10−21 Symbol: z
• Yocto = 10−24 Symbol: y
METRIC PREFIX SCALE
T G M k m µ n p
tera giga mega kilo (none) milli micro nano pico
1012 109 106 103 100 10-3 10-6 10-9 10-12
102 101 10-1 10-2
hecto deca deci centi
h da d c
21.
1.10. METRIC PREFIXES AND UNIT CONVERSIONS 13
• Conversion factors for temperature
• o F = (o C)(9/5) + 32
• o C = (o F - 32)(5/9)
• o R = o F + 459.67
• o K = o C + 273.15
Conversion equivalencies for volume
1 US gallon (gal) = 231.0 cubic inches (in3 ) = 4 quarts (qt) = 8 pints (pt) = 128
ﬂuid ounces (ﬂ. oz.) = 3.7854 liters (l)
1 Imperial gallon (gal) = 160 ﬂuid ounces (ﬂ. oz.) = 4.546 liters (l)
Conversion equivalencies for distance
1 inch (in) = 2.540000 centimeter (cm)
Conversion equivalencies for velocity
1 mile per hour (mi/h) = 88 feet per minute (ft/m) = 1.46667 feet per second (ft/s)
= 1.60934 kilometer per hour (km/h) = 0.44704 meter per second (m/s) = 0.868976
knot (knot – international)
Conversion equivalencies for weight
1 pound (lb) = 16 ounces (oz) = 0.45359 kilogram (kg)
Conversion equivalencies for force
1 pound-force (lbf) = 4.44822 newton (N)
Acceleration of gravity (free fall), Earth standard
9.806650 meters per second per second (m/s2 ) = 32.1740 feet per second per sec-
ond (ft/s2 )
Conversion equivalencies for area
1 acre = 43560 square feet (ft2 ) = 4840 square yards (yd2 ) = 4046.86 square
meters (m2 )
22.
14 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS
Conversion equivalencies for pressure
1 pound per square inch (psi) = 2.03603 inches of mercury (in. Hg) = 27.6807
inches of water (in. W.C.) = 6894.757 pascals (Pa) = 0.0680460 atmospheres (Atm) =
0.0689476 bar (bar)
Conversion equivalencies for energy or work
1 british thermal unit (BTU – ”International Table”) = 251.996 calories (cal –
”International Table”) = 1055.06 joules (J) = 1055.06 watt-seconds (W-s) = 0.293071
watt-hour (W-hr) = 1.05506 x 1010 ergs (erg) = 778.169 foot-pound-force (ft-lbf)
Conversion equivalencies for power
1 horsepower (hp – 550 ft-lbf/s) = 745.7 watts (W) = 2544.43 british thermal units
per hour (BTU/hr) = 0.0760181 boiler horsepower (hp – boiler)
Conversion equivalencies for motor torque
Newton-meter Gram-centimeter Pound-inch Pound-foot Ounce-inch
(n-m) (g-cm) (lb-in) (lb-ft) (oz-in)
n-m 1 1020 8.85 0.738 141.6
-6 -3 -6
g-cm 981 x 10 1 8.68 x 10 723 x 10 0.139
lb-in 0.113 115 1 0.0833 16
lb-ft 1.36 1383 12 1 192
-3 -3
oz-in 7.062 x 10 7.20 0.0625 5.21 x 10 1
Locate the row corresponding to known unit of torque along the left of the table. Multiply
by the factor under the column for the desired units. For example, to convert 2 oz-in torque
to n-m, locate oz-in row at table left. Locate 7.062 x 10−3 at intersection of desired n-m units
column. Multiply 2 oz-in x (7.062 x 10−3 ) = 14.12 x 10−3 n-m.
Converting between units is easy if you have a set of equivalencies to work with. Suppose
we wanted to convert an energy quantity of 2500 calories into watt-hours. What we would need
to do is ﬁnd a set of equivalent ﬁgures for those units. In our reference here, we see that 251.996
calories is physically equal to 0.293071 watt hour. To convert from calories into watt-hours,
we must form a ”unity fraction” with these physically equal ﬁgures (a fraction composed of
different ﬁgures and different units, the numerator and denominator being physically equal to
one another), placing the desired unit in the numerator and the initial unit in the denominator,
and then multiply our initial value of calories by that fraction.
Since both terms of the ”unity fraction” are physically equal to one another, the fraction
as a whole has a physical value of 1, and so does not change the true value of any ﬁgure
when multiplied by it. When units are canceled, however, there will be a change in units.
23.
1.10. METRIC PREFIXES AND UNIT CONVERSIONS 15
For example, 2500 calories multiplied by the unity fraction of (0.293071 w-hr / 251.996 cal) =
2.9075 watt-hours.
Original figure 2500 calories
0.293071 watt-hour
"Unity fraction"
251.996 calories
. . . cancelling units . . .
2500 calories 0.293071 watt-hour
1 251.996 calories
Converted figure 2.9075 watt-hours
The ”unity fraction” approach to unit conversion may be extended beyond single steps. Sup-
pose we wanted to convert a ﬂuid ﬂow measurement of 175 gallons per hour into liters per day.
We have two units to convert here: gallons into liters, and hours into days. Remember that
the word ”per” in mathematics means ”divided by,” so our initial ﬁgure of 175 gallons per hour
means 175 gallons divided by hours. Expressing our original ﬁgure as such a fraction, we
multiply it by the necessary unity fractions to convert gallons to liters (3.7854 liters = 1 gal-
lon), and hours to days (1 day = 24 hours). The units must be arranged in the unity fraction
in such a way that undesired units cancel each other out above and below fraction bars. For
this problem it means using a gallons-to-liters unity fraction of (3.7854 liters / 1 gallon) and a
hours-to-days unity fraction of (24 hours / 1 day):
24.
16 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS
Original figure 175 gallons/hour
3.7854 liters
"Unity fraction"
1 gallon
24 hours
"Unity fraction"
1 day
. . . cancelling units . . .
175 gallons 3.7854 liters 24 hours
1 hour 1 gallon 1 day
Converted figure 15,898.68 liters/day
Our ﬁnal (converted) answer is 15898.68 liters per day.
1.11 Data
Conversion factors were found in the 78th edition of the CRC Handbook of Chemistry and
Physics, and the 3rd edition of Bela Liptak’s Instrument Engineers’ Handbook – Process Mea-
surement and Analysis.
1.12 Contributors
Contributors to this chapter are listed in chronological order of their contributions, from most
recent to ﬁrst. See Appendix 2 (Contributor List) for dates and contact information.
Gerald Gardner (January 2003): Addition of Imperial gallons conversion.
26.
18 CHAPTER 2. COLOR CODES
Color Digit Multiplier Tolerance (%)
Black 0 100 (1)
Brown 1 101 1
2
Red 2 10 2
Orange 3 103
Yellow 4 104
Green 5 105 0.5
6
Blue 6 10 0.25
Violet 7 107 0.1
8
Grey 8 10
White 9 109
Gold 10-1 5
Silver 10-2 10
(none) 20
The colors brown, red, green, blue, and violet are used as tolerance codes on 5-band resistors
only. All 5-band resistors use a colored tolerance band. The blank (20%) ”band” is only used
with the ”4-band” code (3 colored bands + a blank ”band”).
Digit Digit Multiplier Tolerance
4-band code
Digit Digit Digit Multiplier Tolerance
5-band code
27.
2.1. RESISTOR COLOR CODES 19
2.1.1 Example #1
A resistor colored Yellow-Violet-Orange-Gold would be 47 kΩ with a tolerance of +/- 5%.
2.1.2 Example #2
A resistor colored Green-Red-Gold-Silver would be 5.2 Ω with a tolerance of +/- 10%.
2.1.3 Example #3
A resistor colored White-Violet-Black would be 97 Ω with a tolerance of +/- 20%. When you
see only three color bands on a resistor, you know that it is actually a 4-band code with a blank
(20%) tolerance band.
2.1.4 Example #4
A resistor colored Orange-Orange-Black-Brown-Violet would be 3.3 kΩ with a tolerance of
+/- 0.1%.
2.1.5 Example #5
A resistor colored Brown-Green-Grey-Silver-Red would be 1.58 Ω with a tolerance of +/- 2%.
2.1.6 Example #6
A resistor colored Blue-Brown-Green-Silver-Blue would be 6.15 Ω with a tolerance of +/-
0.25%.
28.
20 CHAPTER 2. COLOR CODES
2.2 Wiring Color Codes
Wiring for AC and DC power distribution branch circuits are color coded for identiﬁcation of
individual wires. In some jurisdictions all wire colors are speciﬁed in legal documents. In other
jurisdictions, only a few conductor colors are so codiﬁed. In that case, local custom dictates the
“optional” wire colors.
IEC, AC: Most of Europe abides by IEC (International Electrotechnical Commission) wiring
color codes for AC branch circuits. These are listed in Table 2.1. The older color codes in the
table reﬂect the previous style which did not account for proper phase rotation. The protective
ground wire (listed as green-yellow) is green with yellow stripe.
Table 2.1: IEC (most of Europe) AC power circuit wiring color codes.
Function label Color, IEC Color, old IEC
Protective earth PE green-yellow green-yellow
Neutral N blue blue
Line, single phase L brown brown or black
Line, 3-phase L1 brown brown or black
Line, 3-phase L2 black brown or black
Line, 3-phase L3 grey brown or black
UK, AC: The United Kingdom now follows the IEC AC wiring color codes. Table 2.2 lists
these along with the obsolete domestic color codes. For adding new colored wiring to existing
old colored wiring see Cook. [1]
Table 2.2: UK AC power circuit wiring color codes.
Function label Color, IEC Old UK color
Protective earth PE green-yellow green-yellow
Neutral N blue black
Line, single phase L brown red
Line, 3-phase L1 brown red
Line, 3-phase L2 black yellow
Line, 3-phase L3 grey blue
US, AC:The US National Electrical Code only mandates white (or grey) for the neutral
power conductor and bare copper, green, or green with yellow stripe for the protective ground.
In principle any other colors except these may be used for the power conductors. The colors
adopted as local practice are shown in Table 2.3. Black, red, and blue are used for 208 VAC
three-phase; brown, orange and yellow are used for 480 VAC. Conductors larger than #6 AWG
are only available in black and are color taped at the ends.
Canada: Canadian wiring is governed by the CEC (Canadian Electric Code). See Table 2.4.
The protective ground is green or green with yellow stripe. The neutral is white, the hot (live
or active) single phase wires are black , and red in the case of a second active. Three-phase
lines are red, black, and blue.
29.
2.2. WIRING COLOR CODES 21
Table 2.3: US AC power circuit wiring color codes.
Function label Color, common Color, alternative
Protective ground PG bare, green, or green-yellow green
Neutral N white grey
Line, single phase L black or red (2nd hot)
Line, 3-phase L1 black brown
Line, 3-phase L2 red orange
Line, 3-phase L3 blue yellow
Table 2.4: Canada AC power circuit wiring color codes.
Function label Color, common
Protective ground PG green or green-yellow
Neutral N white
Line, single phase L black or red (2nd hot)
Line, 3-phase L1 red
Line, 3-phase L2 black
Line, 3-phase L3 blue
IEC, DC: DC power installations, for example, solar power and computer data centers, use
color coding which follows the AC standards. The IEC color standard for DC power cables is
listed in Table 2.5, adapted from Table 2, Cook. [1]
Table 2.5: IEC DC power circuit wiring color codes.
Function label Color
Protective earth PE green-yellow
2-wire unearthed DC Power System
Positive L+ brown
Negative L- grey
2-wire earthed DC Power System
Positive (of a negative earthed) circuit L+ brown
Negative (of a negative earthed) circuit M blue
Positive (of a positive earthed) circuit M blue
Negative (of a positive earthed) circuit L- grey
3-wire earthed DC Power System
Positive L+ brown
Mid-wire M blue
Negative L- grey
US DC power: The US National Electrical Code (for both AC and DC) mandates that
the grounded neutral conductor of a power system be white or grey. The protective ground
must be bare, green or green-yellow striped. Hot (active) wires may be any other colors except
these. However, common practice (per local electrical inspectors) is for the ﬁrst hot (live or
active) wire to be black and the second hot to be red. The recommendations in Table 2.6 are
30.
22 CHAPTER 2. COLOR CODES
by Wiles. [2] He makes no recommendation for ungrounded power system colors. Usage of the
ungrounded system is discouraged for safety. However, red (+) and black (-) follows the coloring
of the grounded systems in the table.
Table 2.6: US recommended DC power circuit wiring color codes.
Function label Color
Protective ground PG bare, green, or green-yellow
2-wire ungrounded DC Power System
Positive L+ no recommendation (red)
Negative L- no recommendation (black)
2-wire grounded DC Power System
Positive (of a negative grounded) circuit L+ red
Negative (of a negative grounded) circuit N white
Positive (of a positive grounded) circuit N white
Negative (of a positive grounded) circuit L- black
3-wire grounded DC Power System
Positive L+ red
Mid-wire (center tap) N white
Negative L- black
Bibliography
[1] Paul Cook, “Harmonised colours and alphanumeric marking”, IEE Wiring Matters, Spring
2004 at http://www.iee.org/Publish/WireRegs/IEE Harmonized colours.pdf
[2] John Wiles, “Photovoltaic Power Systems and the National Electrical Code: Suggested
Practices”, Southwest Technology Development Institute, New Mexico State University,
March 2001 at http://www.re.sandia.gov/en/ti/tu/Copy%20of%20NEC2000.pdf
33.
3.3. COEFFICIENTS OF SPECIFIC RESISTANCE 25
| TW RUH THHN, XHHW |
========================================================
Size Current Rating Current Rating Current Rating
AWG @ 60 degrees C @ 75 degrees C @ 90 degrees C
========================================================
20 -------- *9 ----------------------------- *12.5
18 -------- *13 ------------------------------ 18
16 -------- *18 ------------------------------ 24
14 --------- 25 ------------- 30 ------------- 35
12 --------- 30 ------------- 35 ------------- 40
10 --------- 40 ------------- 50 ------------- 55
8 ---------- 60 ------------- 70 ------------- 80
6 ---------- 80 ------------- 95 ------------ 105
4 --------- 105 ------------ 125 ------------ 140
2 --------- 140 ------------ 170 ------------ 190
1 --------- 165 ------------ 195 ------------ 220
1/0 ------- 195 ------------ 230 ------------ 260
2/0 ------- 225 ------------ 265 ------------ 300
3/0 ------- 260 ------------ 310 ------------ 350
4/0 ------- 300 ------------ 360 ------------ 405
* = estimated values; normally, wire gages this small are not manufactured with these
insulation types.
3.3 Coefﬁcients of speciﬁc resistance
Speciﬁc resistance at 20o C:
Material Element/Alloy (ohm-cmil/ft) (ohm-cm·10−6 )
====================================================================
Nichrome ------- Alloy ---------------- 675 ------------- 112.2
Nichrome V ----- Alloy ---------------- 650 ------------- 108.1
Manganin ------- Alloy ---------------- 290 ------------- 48.21
Constantan ----- Alloy ---------------- 272.97 ---------- 45.38
Steel* --------- Alloy ---------------- 100 ------------- 16.62
Platinum ------ Element --------------- 63.16 ----------- 10.5
Iron ---------- Element --------------- 57.81 ----------- 9.61
Nickel -------- Element --------------- 41.69 ----------- 6.93
Zinc ---------- Element --------------- 35.49 ----------- 5.90
Molybdenum ---- Element --------------- 32.12 ----------- 5.34
Tungsten ------ Element --------------- 31.76 ----------- 5.28
Aluminum ------ Element --------------- 15.94 ----------- 2.650
Gold ---------- Element --------------- 13.32 ----------- 2.214
Copper -------- Element --------------- 10.09 ----------- 1.678
Silver -------- Element --------------- 9.546 ----------- 1.587
* = Steel alloy at 99.5 percent iron, 0.5 percent carbon.
34.
26 CHAPTER 3. CONDUCTOR AND INSULATOR TABLES
3.4 Temperature coefﬁcients of resistance
Temperature coefﬁcient (α) per degree C:
Material Element/Alloy Temp. coefficient
=====================================================
Nickel -------- Element --------------- 0.005866
Iron ---------- Element --------------- 0.005671
Molybdenum ---- Element --------------- 0.004579
Tungsten ------ Element --------------- 0.004403
Aluminum ------ Element --------------- 0.004308
Copper -------- Element --------------- 0.004041
Silver -------- Element --------------- 0.003819
Platinum ------ Element --------------- 0.003729
Gold ---------- Element --------------- 0.003715
Zinc ---------- Element --------------- 0.003847
Steel* --------- Alloy ---------------- 0.003
Nichrome ------- Alloy ---------------- 0.00017
Nichrome V ----- Alloy ---------------- 0.00013
Manganin ------- Alloy ------------ +/- 0.000015
Constantan ----- Alloy --------------- -0.000074
* = Steel alloy at 99.5 percent iron, 0.5 percent carbon
3.5 Critical temperatures for superconductors
Critical temperatures given in Kelvins
Material Element/Alloy Critical temperature(K)
=======================================================
Aluminum -------- Element --------------- 1.20
Cadmium --------- Element --------------- 0.56
Lead ------------ Element --------------- 7.2
Mercury --------- Element --------------- 4.16
Niobium --------- Element --------------- 8.70
Thorium --------- Element --------------- 1.37
Tin ------------- Element --------------- 3.72
Titanium -------- Element --------------- 0.39
Uranium --------- ELement --------------- 1.0
Zinc ------------ Element --------------- 0.91
Niobium/Tin ------ Alloy ---------------- 18.1
Cupric sulphide - Compound -------------- 1.6
35.
3.6. DIELECTRIC STRENGTHS FOR INSULATORS 27
Critical temperatures, high temperature superconuctors ni Kelvins
Material Critical temperature(K)
=======================================================
HgBa2 Ca2 Cu3 O8+d ---------------- 150 (23.5 GPa pressure)
HgBa2 Ca2 Cu3 O8+d ---------------- 133
Tl2 Ba2 Ca2 Cu3 O10 ---------------- 125
YBa2 Cu3 O7 ---------------------- 90
La1.85 Sr0.15 CuO4 ----------------- 40
Cs3 C60 ------------------------- 40 (15 Kbar pressure)
Ba0.6 K0.4 BiO3 ------------------- 30
Nd1.85 Ce0.15 CuO4 ----------------- 22
K3 C60 -------------------------- 19
PbMo6 S8 ------------------------ 12.6
Note: all critical temperatures given at zero magnetic ﬁeld strength.
3.6 Dielectric strengths for insulators
Dielectric strength in kilovolts per inch (kV/in):
Material* Dielectric strength
=========================================
Vacuum --------------------- 20
Air ------------------------ 20 to 75
Porcelain ------------------ 40 to 200
Paraffin Wax --------------- 200 to 300
Transformer Oil ------------ 400
Bakelite ------------------- 300 to 550
Rubber --------------------- 450 to 700
Shellac -------------------- 900
Paper ---------------------- 1250
Teflon --------------------- 1500
Glass ---------------------- 2000 to 3000
Mica ----------------------- 5000
* = Materials listed are specially prepared for electrical use
3.7 Data
Tables of speciﬁc resistance and temperature coefﬁcient of resistance for elemental materials
(not alloys) were derived from ﬁgures found in the 78th edition of the CRC Handbook of Chem-
istry and Physics. Superconductivity data from Collier’s Encyclopedia (volume 21, 1968, page
640).
38.
30 CHAPTER 4. ALGEBRA REFERENCE
4.1 Basic identities
a+0=a 1a = a 0a = 0
a =a 0 =0 a =1
1 a a
a = undefined
0
Note: while division by zero is popularly thought to be equal to inﬁnity, this is not techni-
cally true. In some practical applications it may be helpful to think the result of such a fraction
approaching positive inﬁnity as a positive denominator approaches zero (imagine calculating
current I=E/R in a circuit with resistance approaching zero – current would approach inﬁnity),
but the actual fraction of anything divided by zero is undeﬁned in the scope of either real or
complex numbers.
4.2 Arithmetic properties
4.2.1 The associative property
In addition and multiplication, terms may be arbitrarily associated with each other through
the use of parentheses:
a + (b + c) = (a + b) + c a(bc) = (ab)c
4.2.2 The commutative property
In addition and multiplication, terms may be arbitrarily interchanged, or commutated:
a+b=b+a ab=ba
4.2.3 The distributive property
a(b + c) = ab + ac
4.3 Properties of exponents
aman = am+n (ab)m = ambm
am
(am)n = amn = am-n
an
39.
4.4. RADICALS 31
4.4 Radicals
4.4.1 Deﬁnition of a radical
When people talk of a ”square root,” they’re referring to a radical with a root of 2. This is
mathematically equivalent to a number raised to the power of 1/2. This equivalence is useful
to know when using a calculator to determine a strange root. Suppose for example you needed
to ﬁnd the fourth root of a number, but your calculator lacks a ”4th root” button or function. If
it has a yx function (which any scientiﬁc calculator should have), you can ﬁnd the fourth root
by raising that number to the 1/4 power, or x0.25 .
x
a = a1/x
It is important to remember that when solving for an even root (square root, fourth root,
etc.) of any number, there are two valid answers. For example, most people know that the
square root of nine is three, but negative three is also a valid answer, since (-3)2 = 9 just as 32
= 9.
4.4.2 Properties of radicals
x x x
a =a ax = a
x x x
ab = a b
x x
a a
= x
b b
4.5 Important constants
4.5.1 Euler’s number
Euler’s constant is an important value for exponential functions, especially scientiﬁc applica-
tions involving decay (such as the decay of a radioactive substance). It is especially important
in calculus due to its uniquely self-similar properties of integration and differentiation.
e approximately equals:
2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69996
40.
32 CHAPTER 4. ALGEBRA REFERENCE
e= 1
k!
k=0
1 1 1 1 ... 1
0! + 1! + 2! + 3! + n!
4.5.2 Pi
Pi (π) is deﬁned as the ratio of a circle’s circumference to its diameter.
Pi approximately equals:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37511
Note: For both Euler’s constant (e) and pi (π), the spaces shown between each set of ﬁve
digits have no mathematical signiﬁcance. They are placed there just to make it easier for your
eyes to ”piece” the number into ﬁve-digit groups when manually copying.
4.6 Logarithms
4.6.1 Deﬁnition of a logarithm
If:
by = x
Then:
logb x = y
Where,
b = "Base" of the logarithm
”log” denotes a common logarithm (base = 10), while ”ln” denotes a natural logarithm (base
= e).
41.
4.7. FACTORING EQUIVALENCIES 33
4.6.2 Properties of logarithms
(log a) + (log b) = log ab
(log a) - (log b) = log a
b
log am = (m)(log a)
a(log m) = m
These properties of logarithms come in handy for performing complex multiplication and
division operations. They are an example of something called a transform function, whereby
one type of mathematical operation is transformed into another type of mathematical operation
that is simpler to solve. Using a table of logarithm ﬁgures, one can multiply or divide numbers
by adding or subtracting their logarithms, respectively. then looking up that logarithm ﬁgure
in the table and seeing what the ﬁnal product or quotient is.
Slide rules work on this principle of logarithms by performing multiplication and division
through addition and subtraction of distances on the slide.
Slide rule
Cursor
Slide
Numerical quantities are represented by
the positioning of the slide.
Marks on a slide rule’s scales are spaced in a logarithmic fashion, so that a linear posi-
tioning of the scale or cursor results in a nonlinear indication as read on the scale(s). Adding
or subtracting lengths on these logarithmic scales results in an indication equivalent to the
product or quotient, respectively, of those lengths.
Most slide rules were also equipped with special scales for trigonometric functions, powers,
roots, and other useful arithmetic functions.
4.7 Factoring equivalencies
x2 - y2 = (x+y)(x-y)
x3 - y3 = (x-y)(x2 + xy + y2)
42.
34 CHAPTER 4. ALGEBRA REFERENCE
4.8 The quadratic formula
For a polynomial expression in
the form of: ax2 + bx + c = 0
+
x = -b - b2 - 4ac
2a
4.9 Sequences
4.9.1 Arithmetic sequences
An arithmetic sequence is a series of numbers obtained by adding (or subtracting) the same
value with each step. A child’s counting sequence (1, 2, 3, 4, . . .) is a simple arithmetic
sequence, where the common difference is 1: that is, each adjacent number in the sequence
differs by a value of one. An arithmetic sequence counting only even numbers (2, 4, 6, 8, . . .)
or only odd numbers (1, 3, 5, 7, 9, . . .) would have a common difference of 2.
In the standard notation of sequences, a lower-case letter ”a” represents an element (a
single number) in the sequence. The term ”an ” refers to the element at the nth step in the
sequence. For example, ”a3 ” in an even-counting (common difference = 2) arithmetic sequence
starting at 2 would be the number 6, ”a” representing 4 and ”a1 ” representing the starting
point of the sequence (given in this example as 2).
A capital letter ”A” represents the sum of an arithmetic sequence. For instance, in the same
even-counting sequence starting at 2, A4 is equal to the sum of all elements from a1 through
a4 , which of course would be 2 + 4 + 6 + 8, or 20.
an = an-1 + d an = a1 + d(n-1)
Where:
d = The "common difference"
Example of an arithmetic sequence:
-7, -3, 1, 5, 9, 13, 17, 21, 25 . . .
An = a1 + a2 + . . . an
An = n (a1 + an)
2
43.
4.10. FACTORIALS 35
4.9.2 Geometric sequences
A geometric sequence, on the other hand, is a series of numbers obtained by multiplying (or
dividing) by the same value with each step. A binary place-weight sequence (1, 2, 4, 8, 16, 32,
64, . . .) is a simple geometric sequence, where the common ratio is 2: that is, each adjacent
number in the sequence differs by a factor of two.
an = r(an-1) an = a1(rn-1)
Where:
r = The "common ratio"
Example of a geometric sequence:
3, 12, 48, 192, 768, 3072 . . .
An = a1 + a2 + . . . an
a1(1 - rn)
An =
1-r
4.10 Factorials
4.10.1 Deﬁnition of a factorial
Denoted by the symbol ”!” after an integer; the product of that integer and all integers in
descent to 1.
Example of a factorial:
5! = 5 x 4 x 3 x 2 x 1
5! = 120
4.10.2 Strange factorials
0! = 1 1! = 1
4.11 Solving simultaneous equations
The terms simultaneous equations and systems of equations refer to conditions where two or
more unknown variables are related to each other through an equal number of equations.
Consider the following example:
44.
36 CHAPTER 4. ALGEBRA REFERENCE
x + y = 24
2x - y = -6
For this set of equations, there is but a single combination of values for x and y that will
satisfy both. Either equation, considered separately, has an inﬁnitude of valid (x,y) solutions,
but together there is only one. Plotted on a graph, this condition becomes obvious:
(6,18)
x + y = 24
2x - y = -6
Each line is actually a continuum of points representing possible x and y solution pairs for
each equation. Each equation, separately, has an inﬁnite number of ordered pair (x,y) solu-
tions. There is only one point where the two linear functions x + y = 24 and 2x - y = -6
intersect (where one of their many independent solutions happen to work for both equations),
and that is where x is equal to a value of 6 and y is equal to a value of 18.
Usually, though, graphing is not a very efﬁcient way to determine the simultaneous solution
set for two or more equations. It is especially impractical for systems of three or more variables.
In a three-variable system, for example, the solution would be found by the point intersection
of three planes in a three-dimensional coordinate space – not an easy scenario to visualize.
4.11.1 Substitution method
Several algebraic techniques exist to solve simultaneous equations. Perhaps the easiest to
comprehend is the substitution method. Take, for instance, our two-variable example problem:
x + y = 24
2x - y = -6
In the substitution method, we manipulate one of the equations such that one variable is
deﬁned in terms of the other:
45.
4.11. SOLVING SIMULTANEOUS EQUATIONS 37
x + y = 24
y = 24 - x
Defining y in terms of x
Then, we take this new deﬁnition of one variable and substitute it for the same variable in
the other equation. In this case, we take the deﬁnition of y, which is 24 - x and substitute
this for the y term found in the other equation:
y = 24 - x
substitute
2x - y = -6
2x - (24 - x) = -6
Now that we have an equation with just a single variable (x), we can solve it using ”normal”
algebraic techniques:
2x - (24 - x) = -6
Distributive property
2x - 24 + x = -6
Combining like terms
3x -24 = -6
Adding 24 to each side
3x = 18
Dividing both sides by 3
x=6
Now that x is known, we can plug this value into any of the original equations and obtain
a value for y. Or, to save us some work, we can plug this value (6) into the equation we just
generated to deﬁne y in terms of x, being that it is already in a form to solve for y:
46.
38 CHAPTER 4. ALGEBRA REFERENCE
x=6
substitute
y = 24 - x
y = 24 - 6
y = 18
Applying the substitution method to systems of three or more variables involves a similar
pattern, only with more work involved. This is generally true for any method of solution:
the number of steps required for obtaining solutions increases rapidly with each additional
variable in the system.
To solve for three unknown variables, we need at least three equations. Consider this
example:
x - y + z = 10
3x + y + 2z = 34
-5x + 2y - z = -14
Being that the ﬁrst equation has the simplest coefﬁcients (1, -1, and 1, for x, y, and z,
respectively), it seems logical to use it to develop a deﬁnition of one variable in terms of the
other two. In this example, I’ll solve for x in terms of y and z:
x - y + z = 10
Adding y and subtracting z
from both sides
x = y - z + 10
Now, we can substitute this deﬁnition of x where x appears in the other two equations:
x = y - z + 10 x = y - z + 10
substitute substitute
3x + y + 2z = 34 -5x + 2y - z = -14
3(y - z + 10) + y + 2z = 34 -5(y - z + 10) + 2y - z = -14
Reducing these two equations to their simplest forms:
47.
4.11. SOLVING SIMULTANEOUS EQUATIONS 39
3(y - z + 10) + y + 2z = 34 -5(y - z + 10) + 2y - z = -14
Distributive property
3y - 3z + 30 + y + 2z = 34 -5y + 5z - 50 + 2y - z = -14
Combining like terms
4y - z + 30 = 34 -3y + 4z - 50 = -14
Moving constant values to right
of the "=" sign
4y - z = 4 -3y + 4z = 36
So far, our efforts have reduced the system from three variables in three equations to two
variables in two equations. Now, we can apply the substitution technique again to the two
equations 4y - z = 4 and -3y + 4z = 36 to solve for either y or z. First, I’ll manipulate
the ﬁrst equation to deﬁne z in terms of y:
4y - z = 4
Adding z to both sides;
subtracting 4 from both sides
z = 4y - 4
Next, we’ll substitute this deﬁnition of z in terms of y where we see z in the other equation:
z = 4y - 4
substitute
-3y + 4z = 36
-3y + 4(4y - 4) = 36
Distributive property
-3y + 16y - 16 = 36
Combining like terms
13y - 16 = 36
Adding 16 to both sides
13y = 52
Dividing both sides by 13
y=4
Now that y is a known value, we can plug it into the equation deﬁning z in terms of y and
48.
40 CHAPTER 4. ALGEBRA REFERENCE
obtain a ﬁgure for z:
y=4
substitute
z = 4y - 4
z = 16 - 4
z = 12
Now, with values for y and z known, we can plug these into the equation where we deﬁned
x in terms of y and z, to obtain a value for x:
y=4
substitute z = 12
substitute
x = y - z + 10
x = 4 - 12 + 10
x=2
In closing, we’ve found values for x, y, and z of 2, 4, and 12, respectively, that satisfy all
three equations.
4.11.2 Addition method
While the substitution method may be the easiest to grasp on a conceptual level, there are
other methods of solution available to us. One such method is the so-called addition method,
whereby equations are added to one another for the purpose of canceling variable terms.
Let’s take our two-variable system used to demonstrate the substitution method:
x + y = 24
2x - y = -6
One of the most-used rules of algebra is that you may perform any arithmetic operation you
wish to an equation so long as you do it equally to both sides. With reference to addition, this
means we may add any quantity we wish to both sides of an equation – so long as its the same
quantity – without altering the truth of the equation.
An option we have, then, is to add the corresponding sides of the equations together to form
a new equation. Since each equation is an expression of equality (the same quantity on either
49.
4.11. SOLVING SIMULTANEOUS EQUATIONS 41
side of the = sign), adding the left-hand side of one equation to the left-hand side of the other
equation is valid so long as we add the two equations’ right-hand sides together as well. In our
example equation set, for instance, we may add x + y to 2x - y, and add 24 and -6 together
as well to form a new equation. What beneﬁt does this hold for us? Examine what happens
when we do this to our example equation set:
x + y = 24
+ 2x - y = -6
3x + 0 = 18
Because the top equation happened to contain a positive y term while the bottom equation
happened to contain a negative y term, these two terms canceled each other in the process of
addition, leaving no y term in the sum. What we have left is a new equation, but one with only
a single unknown variable, x! This allows us to easily solve for the value of x:
3x + 0 = 18
Dropping the 0 term
3x = 18
Dividing both sides by 3
x=6
Once we have a known value for x, of course, determining y’s value is a simply matter of
substitution (replacing x with the number 6) into one of the original equations. In this example,
the technique of adding the equations together worked well to produce an equation with a
single unknown variable. What about an example where things aren’t so simple? Consider the
following equation set:
2x + 2y = 14
3x + y = 13
We could add these two equations together – this being a completely valid algebraic opera-
tion – but it would not proﬁt us in the goal of obtaining values for x and y:
2x + 2y = 14
+ 3x + y = 13
5x + 3y = 27
The resulting equation still contains two unknown variables, just like the original equations
do, and so we’re no further along in obtaining a solution. However, what if we could manipulate
one of the equations so as to have a negative term that would cancel the respective term in the
other equation when added? Then, the system would reduce to a single equation with a single
unknown variable just as with the last (fortuitous) example.
If we could only turn the y term in the lower equation into a - 2y term, so that when the
two equations were added together, both y terms in the equations would cancel, leaving us
with only an x term, this would bring us closer to a solution. Fortunately, this is not difﬁcult to
do. If we multiply each and every term of the lower equation by a -2, it will produce the result
50.
42 CHAPTER 4. ALGEBRA REFERENCE
we seek:
-2(3x + y) = -2(13)
Distributive property
-6x - 2y = -26
Now, we may add this new equation to the original, upper equation:
2x + 2y = 14
+ -6x - 2y = -26
-4x + 0y = -12
Solving for x, we obtain a value of 3:
-4x + 0y = -12
Dropping the 0 term
-4x = -12
Dividing both sides by -4
x=3
Substituting this new-found value for x into one of the original equations, the value of y is
easily determined:
x=3
substitute
2x + 2y = 14
6 + 2y = 14
Subtracting 6 from both sides
2y = 8
Dividing both sides by 2
y=4
Using this solution technique on a three-variable system is a bit more complex. As with
substitution, you must use this technique to reduce the three-equation system of three vari-
ables down to two equations with two variables, then apply it again to obtain a single equation
with one unknown variable. To demonstrate, I’ll use the three-variable equation system from
the substitution section:
51.
4.11. SOLVING SIMULTANEOUS EQUATIONS 43
x - y + z = 10
3x + y + 2z = 34
-5x + 2y - z = -14
Being that the top equation has coefﬁcient values of 1 for each variable, it will be an easy
equation to manipulate and use as a cancellation tool. For instance, if we wish to cancel the 3x
term from the middle equation, all we need to do is take the top equation, multiply each of its
terms by -3, then add it to the middle equation like this:
x - y + z = 10
Multiply both sides by -3
-3(x - y + z) = -3(10)
Distributive property
-3x + 3y - 3z = -30
-3x + 3y - 3z = -30
(Adding)
+ 3x + y + 2z = 34
0x + 4y - z = 4
or
4y - z = 4
We can rid the bottom equation of its -5x term in the same manner: take the original
top equation, multiply each of its terms by 5, then add that modiﬁed equation to the bottom
equation, leaving a new equation with only y and z terms:
52.
44 CHAPTER 4. ALGEBRA REFERENCE
x - y + z = 10
Multiply both sides by 5
5(x - y + z) = 5(10)
Distributive property
5x - 5y + 5z = 50
5x - 5y + 5z = 50
(Adding)
+ -5x + 2y - z = -14
0x - 3y + 4z = 36
or
-3y + 4z = 36
At this point, we have two equations with the same two unknown variables, y and z:
4y - z = 4
-3y + 4z = 36
By inspection, it should be evident that the -z term of the upper equation could be leveraged
to cancel the 4z term in the lower equation if only we multiply each term of the upper equation
by 4 and add the two equations together:
4y - z = 4
Multiply both sides by 4
4(4y - z) = 4(4)
Distributive property
16y - 4z = 16
16y - 4z = 16
(Adding)
+ -3y + 4z = 36
13y + 0z = 52
or
13y = 52
Taking the new equation 13y = 52 and solving for y (by dividing both sides by 13), we get
a value of 4 for y. Substituting this value of 4 for y in either of the two-variable equations
53.
4.12. CONTRIBUTORS 45
allows us to solve for z. Substituting both values of y and z into any one of the original, three-
variable equations allows us to solve for x. The ﬁnal result (I’ll spare you the algebraic steps,
since you should be familiar with them by now!) is that x = 2, y = 4, and z = 12.
4.12 Contributors
Contributors to this chapter are listed in chronological order of their contributions, from most
recent to ﬁrst. See Appendix 2 (Contributor List) for dates and contact information.
Chirvasuta Constantin (April 2, 2003): Pointed out error in quadratic equation formula.
56.
48 CHAPTER 5. TRIGONOMETRY REFERENCE
5.1.1 Trigonometric identities
sin x = O cos x = A tan x = O sin x
tan x = cos x
H H A
csc x = H sec x = H cot x = A cot x = cos x
sin x
O A O
H is the Hypotenuse, always being opposite the right angle. Relative to angle x, O is the
Opposite and A is the Adjacent.
”Arc” functions such as ”arcsin”, ”arccos”, and ”arctan” are the complements of normal
trigonometric functions. These functions return an angle for a ratio input. For example, if
the tangent of 45o is equal to 1, then the ”arctangent” (arctan) of 1 is 45o . ”Arc” functions are
useful for ﬁnding angles in a right triangle if the side lengths are known.
5.1.2 The Pythagorean theorem
H2 = A2 + O2
5.2 Non-right triangle trigonometry
b
C A
a c
B
5.2.1 The Law of Sines (for any triangle)
sin a = sin b = sin c
A B C
57.
5.3. TRIGONOMETRIC EQUIVALENCIES 49
5.2.2 The Law of Cosines (for any triangle)
A2 = B2 + C2 - (2BC)(cos a)
B2 = A2 + C2 - (2AC)(cos b)
C2 = A2 + B2 - (2AB)(cos c)
5.3 Trigonometric equivalencies
sin -x = -sin x cos -x = cos x tan -t = -tan t
csc -t = -csc t sec -t = sec t cot -t = -cot t
sin 2x = 2(sin x)(cos x) cos 2x = (cos2 x) - (sin2 x)
2(tan x)
tan 2t =
1 - tan2 x
sin2 x = 1 - cos 2x cos2 x = 1 + cos 2x
2 2 2 2
5.4 Hyperbolic functions
ex - e-x
sinh x =
2
ex + e-x
cosh x =
2
sinh x
tanh x =
cosh x
Note: all angles (x) must be expressed in units of radians for these hyperbolic functions.
There are 2π radians in a circle (360o ).
5.5 Contributors
Contributors to this chapter are listed in chronological order of their contributions, from most
recent to ﬁrst. See Appendix 2 (Contributor List) for dates and contact information.
58.
50 CHAPTER 5. TRIGONOMETRY REFERENCE
Harvey Lew (??? 2003): Corrected typographical error: ”tangent” should have been ”cotan-
gent”.
60.
52 CHAPTER 6. CALCULUS REFERENCE
6.1 Rules for limits
lim [f(x) + g(x)] = lim f(x) + lim g(x)
x→a x→a x→a
lim [f(x) - g(x)] = lim f(x) - lim g(x)
x→a x→a x→a
lim [f(x) g(x)] = [lim f(x)] [lim g(x)]
x→a x→a x→a
6.2 Derivative of a constant
If:
f(x) = c
Then:
d f(x) = 0
dx
(”c” being a constant)
6.3 Common derivatives
d xn = nxn-1
dx
d ln x = 1
dx x
d ax = (ln a)(ax)
dx
6.4 Derivatives of power functions of e
If: If:
x
f(x) = e f(x) = eg(x)
Then: Then:
d f(x) = ex d f(x) = eg(x) d g(x)
dx dx dx
61.
6.5. TRIGONOMETRIC DERIVATIVES 53
Example:
2
f(x) = e(x + 2x)
d f(x) = e(x2 + 2x) d (x2 + 2x)
dx dx
d f(x) = (e(x2 + 2x))(2x + 2)
dx
6.5 Trigonometric derivatives
d sin x = cos x d cos x = -sin x
dx dx
d tan x = sec2 x d cot x = -csc2 x
dx dx
d sec x = (sec x)(tan x) d csc x = (-csc x)(cot x)
dx dx
6.6 Rules for derivatives
6.6.1 Constant rule
d [cf(x)] = c d f(x)
dx dx
6.6.2 Rule of sums
d [f(x) + g(x)] = d f(x) + d g(x)
dx dx dx
6.6.3 Rule of differences
d [f(x) - g(x)] = d f(x) - d g(x)
dx dx dx
62.
54 CHAPTER 6. CALCULUS REFERENCE
6.6.4 Product rule
d [f(x) g(x)] = f(x)[ d g(x)] + g(x)[ d f(x)]
dx dx dx
6.6.5 Quotient rule
f(x) g(x)[ d f(x)] - f(x)[ d g(x)]
d = dx dx
dx g(x) [g(x)] 2
6.6.6 Power rule
d f(x)a = a[f(x)]a-1 d f(x)
dx dx
6.6.7 Functions of other functions
d f[g(x)]
dx
Break the function into two functions:
u = g(x) and y = f(u)
Solve:
dy f[g(x)] = dy f(u) du g(x)
dx du dx
63.
6.7. THE ANTIDERIVATIVE (INDEFINITE INTEGRAL) 55
6.7 The antiderivative (Indeﬁnite integral)
If:
d f(x) = g(x)
dx
Then:
g(x) is the derivative of f(x)
f(x) is the antiderivative of g(x)
∫g(x) dx = f(x) + c
Notice something important here: taking the derivative of f(x) may precisely give you g(x),
but taking the antiderivative of g(x) does not necessarily give you f(x) in its original form.
Example:
f(x) = 3x2 + 5
d f(x) = 6x
dx
∫6x dx = 3x2 + c
Note that the constant c is unknown! The original function f(x) could have been 3x2 + 5,
3x + 10, 3x2 + anything, and the derivative of f(x) would have still been 6x. Determining the
2
antiderivative of a function, then, is a bit less certain than determining the derivative of a
function.
6.8 Common antiderivatives
∫xn dx = xn+1 + c
n+1
∫ 1 dx = (ln |x|) + c
x
Where,
c = a constant
∫ax dx = ax + c
ln a
64.
56 CHAPTER 6. CALCULUS REFERENCE
6.9 Antiderivatives of power functions of e
∫ex dx = ex + c
Note: this is a very unique and useful property of e. As in the case of derivatives, the
antiderivative of such a function is that same function. In the case of the antiderivative, a
constant term ”c” is added to the end as well.
6.10 Rules for antiderivatives
6.10.1 Constant rule
∫cf(x) dx = c ∫f(x) dx
6.10.2 Rule of sums
∫[f(x) + g(x)] dx = [∫f(x) dx ] + [∫g(x) dx ]
6.10.3 Rule of differences
∫[f(x) - g(x)] dx = [∫f(x) dx ] - [∫g(x) dx ]
6.11 Deﬁnite integrals and the fundamental theorem of
calculus
If:
∫f(x) dx = g(x) or d g(x) = f(x)
dx
Then:
b
∫f(x) dx = g(b) - g(a)
a
Where,
a and b are constants
65.
6.12. DIFFERENTIAL EQUATIONS 57
If:
∫f(x) dx = g(x) and a=0
Then:
x
∫f(x) dx = g(x)
0
6.12 Differential equations
As opposed to normal equations where the solution is a number, a differential equation is one
where the solution is actually a function, and which at least one derivative of that unknown
function is part of the equation.
As with ﬁnding antiderivatives of a function, we are often left with a solution that encom-
passes more than one possibility (consider the many possible values of the constant ”c” typically
found in antiderivatives). The set of functions which answer any differential equation is called
the ”general solution” for that differential equation. Any one function out of that set is re-
ferred to as a ”particular solution” for that differential equation. The variable of reference for
differentiation and integration within the differential equation is known as the ”independent
variable.”
68.
60 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
7.8.8 AC phase shift demonstration . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.8.9 Transformer circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.8.10 Full-wave bridge rectiﬁer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.8.11 Common-base BJT transistor ampliﬁer . . . . . . . . . . . . . . . . . . . 99
7.8.12 Common-source JFET ampliﬁer with self-bias . . . . . . . . . . . . . . . 102
7.8.13 Inverting op-amp circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.8.14 Noninverting op-amp circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.8.15 Instrumentation ampliﬁer . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.8.16 Op-amp integrator with sinewave input . . . . . . . . . . . . . . . . . . . 108
7.8.17 Op-amp integrator with squarewave input . . . . . . . . . . . . . . . . . . 110
7.1 Introduction
”With Electronics Workbench, you can create circuit schematics that look just the
same as those you’re already familiar with on paper – plus you can ﬂip the power
switch so the schematic behaves like a real circuit. With other electronics simulators,
you may have to type in SPICE node lists as text ﬁles – an abstract representation of
a circuit beyond the capabilities of all but advanced electronics engineers.”
(Electronics Workbench User’s guide – version 4, page 7)
This introduction comes from the operating manual for a circuit simulation program called
Electronics Workbench. Using a graphic interface, it allows the user to draw a circuit schematic
and then have the computer analyze that circuit, displaying the results in graphic form. It is a
very valuable analysis tool, but it has its shortcomings. For one, it and other graphic programs
like it tend to be unreliable when analyzing complex circuits, as the translation from picture
to computer code is not quite the exact science we would want it to be (yet). Secondly, due to its
graphics requirements, it tends to need a signiﬁcant amount of computational ”horsepower” to
run, and a computer operating system that supports graphics. Thirdly, these graphic programs
can be costly.
However, underneath the graphics skin of Electronics Workbench lies a robust (and free!)
program called SPICE, which analyzes a circuit based on a text-ﬁle description of the circuit’s
components and connections. What the user pays for with Electronics Workbench and other
graphic circuit analysis programs is the convenient ”point and click” interface, while SPICE
does the actual mathematical analysis.
By itself, SPICE does not require a graphic interface and demands little in system re-
sources. It is also very reliable. The makers of Electronic Workbench would like you to think
that using SPICE in its native text mode is a task suited for rocket scientists, but I’m writing
this to prove them wrong. SPICE is fairly easy to use for simple circuits, and its non-graphic
interface actually lends itself toward the analysis of circuits that can be difﬁcult to draw. I
think it was the programming expert Donald Knuth who quipped, ”What you see is all you get”
when it comes to computer applications. Graphics may look more attractive, but abstracted
interfaces (text) are actually more efﬁcient.
69.
7.2. HISTORY OF SPICE 61
This document is not intended to be an exhaustive tutorial on how to use SPICE. I’m merely
trying to show the interested user how to apply it to the analysis of simple circuits, as an
alternative to proprietary ($$$) and buggy programs. Once you learn the basics, there are
other tutorials better suited to take you further. Using SPICE – a program originally intended
to develop integrated circuits – to analyze some of the really simple circuits showcased here
may seem a bit like cutting butter with a chain saw, but it works!
All options and examples have been tested on SPICE version 2g6 on both MS-DOS and
Linux operating systems. As far as I know, I’m not using features speciﬁc to version 2g6, so
these simple functions should work on most versions of SPICE.
7.2 History of SPICE
SPICE is a computer program designed to simulate analog electronic circuits. It original intent
was for the development of integrated circuits, from which it derived its name: Simulation
Program with Integrated Circuit Emphasis.
The origin of SPICE traces back to another circuit simulation program called CANCER.
Developed by professor Ronald Rohrer of U.C. Berkeley along with some of his students in the
late 1960’s, CANCER continued to be improved through the early 1970’s. When Rohrer left
Berkeley, CANCER was re-written and re-named to SPICE, released as version 1 to the public
domain in May of 1972. Version 2 of SPICE was released in 1975 (version 2g6 – the version
used in this book – is a minor revision of this 1975 release). Instrumental in the decision
to release SPICE as a public-domain computer program was professor Donald Pederson of
Berkeley, who believed that all signiﬁcant technical progress happens when information is
freely shared. I for one thank him for his vision.
A major improvement came about in March of 1985 with version 3 of SPICE (also released
under public domain). Written in the C language rather than FORTRAN, version 3 incorpo-
rated additional transistor types (the MESFET, for example), and switch elements. Version 3
also allowed the use of alphabetical node labels rather than only numbers. Instructions written
for version 2 of SPICE should still run in version 3, though.
Despite the additional power of version 3, I have chosen to use version 2g6 throughout
this book because it seems to be the easiest version to acquire and run on different computer
systems.
7.3 Fundamentals of SPICE programming
Programming a circuit simulation with SPICE is much like programming in any other com-
puter language: you must type the commands as text in a ﬁle, save that ﬁle to the computer’s
hard drive, and then process the contents of that ﬁle with a program (compiler or interpreter)
that understands such commands.
In an interpreted computer language, the computer holds a special program called an inter-
preter that translates the program you wrote (the so-called source ﬁle) into the computer’s own
language, on the ﬂy, as its being executed:
70.
62 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
Computer
Source Interpreter
software Output
File
In a compiled computer language, the program you wrote is translated all at once into the
computer’s own language by a special program called a compiler. After the program you’ve
written has been ”compiled,” the resulting executable ﬁle needs no further translation to be un-
derstood directly by the computer. It can now be ”run” on a computer whether or not compiler
software has been installed on that computer:
Computer
Source Compiler
File software
Computer
Executable Executable
File File
Output
SPICE is an interpreted language. In order for a computer to be able to understand the
SPICE instructions you type, it must have the SPICE program (interpreter) installed:
Computer
Source
File SPICE
software Output
"netlist"
SPICE source ﬁles are commonly referred to as ”netlists,” although they are sometimes
known as ”decks” with each line in the ﬁle being called a ”card.” Cute, don’t you think? Netlists
are created by a person like yourself typing instructions line-by-line using a word processor
or text editor. Text editors are much preferred over word processors for any type of computer
programming, as they produce pure ASCII text with no special embedded codes for text high-
71.
7.3. FUNDAMENTALS OF SPICE PROGRAMMING 63
lighting (like italic or boldface fonts), which are uninterpretable by interpreter and compiler
software.
As in general programming, the source ﬁle you create for SPICE must follow certain con-
ventions of programming. It is a computer language in itself, albeit a simple one. Having
programmed in BASIC and C/C++, and having some experience reading PASCAL and FOR-
TRAN programs, it is my opinion that the language of SPICE is much simpler than any of
these. It is about the same complexity as a markup language such as HTML, perhaps less so.
There is a cycle of steps to be followed in using SPICE to analyze a circuit. The cycle starts
when you ﬁrst invoke the text editing program and make your ﬁrst draft of the netlist. The
next step is to run SPICE on that new netlist and see what the results are. If you are a novice
user of SPICE, your ﬁrst attempts at creating a good netlist will be fraught with small errors
of syntax. Don’t worry – as every computer programmer knows, proﬁciency comes with lots of
practice. If your trial run produces error messages or results that are obviously incorrect, you
need to re-invoke the text editing program and modify the netlist. After modifying the netlist,
you need to run SPICE again and check the results. The sequence, then, looks something like
this:
• Compose a new netlist with a text editing program. Save that netlist to a ﬁle with a name
of your choice.
• Run SPICE on that netlist and observe the results.
• If the results contain errors, start up the text editing program again and modify the
netlist.
• Run SPICE again and observe the new results.
• If there are still errors in the output of SPICE, re-edit the netlist again with the text
editing program. Repeat this cycle of edit/run as many times as necessary until you are
getting the desired results.
• Once you’ve ”debugged” your netlist and are getting good results, run SPICE again, only
this time redirecting the output to a new ﬁle instead of just observing it on the computer
screen.
• Start up a text editing program or a word processor program and open the SPICE output
ﬁle you just created. Modify that ﬁle to suit your formatting needs and either save those
changes to disk and/or print them out on paper.
To ”run” a SPICE ”program,” you need to type in a command at a terminal prompt interface,
such as that found in MS-DOS, UNIX, or the MS-Windows DOS prompt option:
spice < example.cir
The word ”spice” invokes the SPICE interpreting program (providing that the SPICE soft-
ware has been installed on the computer!), the ”<” symbol redirects the contents of the source
ﬁle to the SPICE interpreter, and example.cir is the name of the source ﬁle for this circuit
example. The ﬁle extension ”.cir” is not mandatory; I have seen ”.inp” (for ”input”) and just
72.
64 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
plain ”.txt” work well, too. It will even work when the netlist ﬁle has no extension. SPICE
doesn’t care what you name it, so long as it has a name compatible with the ﬁlesystem of
your computer (for old MS-DOS machines, for example, the ﬁlename must be no more than 8
characters in length, with a 3 character extension, and no spaces or other non-alphanumerical
characters).
When this command is typed in, SPICE will read the contents of the example.cir ﬁle,
analyze the circuit speciﬁed by that ﬁle, and send a text report to the computer terminal’s
standard output (usually the screen, where you can see it scroll by). A typical SPICE out-
put is several screens worth of information, so you might want to look it over with a slight
modiﬁcation of the command:
spice < example.cir | more
This alternative ”pipes” the text output of SPICE to the ”more” utility, which allows only one
page to be displayed at a time. What this means (in English) is that the text output of SPICE
is halted after one screen-full, and waits until the user presses a keyboard key to display the
next screen-full of text. If you’re just testing your example circuit ﬁle and want to check for
any errors, this is a good way to do it.
spice < example.cir > example.txt
This second alternative (above) redirects the text output of SPICE to another ﬁle, called
example.txt, where it can be viewed or printed. This option corresponds to the last step
in the development cycle listed earlier. It is recommended by this author that you use this
technique of ”redirection” to a text ﬁle only after you’ve proven your example circuit netlist to
work well, so that you don’t waste time invoking a text editor just to see the output during the
stages of ”debugging.”
Once you have a SPICE output stored in a .txt ﬁle, you can use a text editor or (better
yet!) a word processor to edit the output, deleting any unnecessary banners and messages,
even specifying alternative fonts to highlight the headings and/or data for a more polished
appearance. Then, of course, you can print the output to paper if you so desire. Being that the
direct SPICE output is plain ASCII text, such a ﬁle will be universally interpretable on any
computer whether SPICE is installed on it or not. Also, the plain text format ensures that the
ﬁle will be very small compared to the graphic screen-shot ﬁles generated by ”point-and-click”
simulators.
The netlist ﬁle format required by SPICE is quite simple. A netlist ﬁle is nothing more
than a plain ASCII text ﬁle containing multiple lines of text, each line describing either a
circuit component or special SPICE command. Circuit architecture is speciﬁed by assigning
numbers to each component’s connection points in each line, connections between components
designated by common numbers. Examine the following example circuit diagram and its cor-
responding SPICE ﬁle. Please bear in mind that the circuit diagram exists only to make the
simulation easier for human beings to understand. SPICE only understands netlists:
73.
7.3. FUNDAMENTALS OF SPICE PROGRAMMING 65
1 1 R2
2
3.3 kΩ
15 V R1 2.2 kΩ R3
150 Ω
0 0 0
Example netlist
v1 1 0 dc 15
r1 1 0 2.2k
r2 1 2 3.3k
r3 2 0 150
.end
Each line of the source ﬁle shown above is explained here:
• v1 represents the battery (voltage source 1), positive terminal numbered 1, negative ter-
minal numbered 0, with a DC voltage output of 15 volts.
• r1 represents resistor R1 in the diagram, connected between points 1 and 0, with a value
of 2.2 kΩ.
• r2 represents resistor R2 in the diagram, connected between points 1 and 2, with a value
of 3.3 kΩ.
• r3 represents resistor R3 in the diagram, connected between points 2 and 0, with a value
of 150 kΩ.
Electrically common points (or ”nodes”) in a SPICE circuit description share common num-
bers, much in the same way that wires connecting common points in a large circuit typically
share common wire labels.
To simulate this circuit, the user would type those six lines of text on a text editor and
save them as a ﬁle with a unique name (such as example.cir). Once the netlist is composed
and saved to a ﬁle, the user then processes that ﬁle with one of the command-line statements
shown earlier (spice < example.cir), and will receive this text output on their computer’s
screen:
1*******10/10/99 ******** spice 2g.6 3/15/83 ********07:32:42*****
0example netlist
0**** input listing temperature = 27.000 deg c
v1 1 0 dc 15
r1 1 0 2.2k
r2 1 2 3.3k
r3 2 0 150
.end
74.
66 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
*****10/10/99 ********* spice 2g.6 3/15/83 ******07:32:42******
0example netlist
0**** small signal bias solution temperature = 27.000 deg c
node voltage node voltage
( 1) 15.0000 ( 2) 0.6522
voltage source currents
name current
v1 -1.117E-02
total power dissipation 1.67E-01 watts
job concluded
0 total job time 0.02
1*******10/10/99 ******** spice 2g.6 3/15/83 ******07:32:42*****
0**** input listing temperature = 27.000 deg c
SPICE begins by printing the time, date, and version used at the top of the output. It then
lists the input parameters (the lines of the source ﬁle), followed by a display of DC voltage
readings from each node (reference number) to ground (always reference number 0). This is
followed by a list of current readings through each voltage source (in this case there’s only one,
v1). Finally, the total power dissipation and computation time in seconds is printed.
All output values provided by SPICE are displayed in scientiﬁc notation.
The SPICE output listing shown above is a little verbose for most peoples’ taste. For a ﬁnal
presentation, it might be nice to trim all the unnecessary text and leave only what matters.
Here is a sample of that same output, redirected to a text ﬁle (spice < example.cir >
example.txt), then trimmed down judiciously with a text editor for ﬁnal presentation and
printed:
example netlist
v1 1 0 dc 15
r1 1 0 2.2k
r2 1 2 3.3k
r3 2 0 150
.end
node voltage node voltage
( 1) 15.0000 ( 2) 0.6522
voltage source currents
name current
v1 -1.117E-02
total power dissipation 1.67E-01 watts
One of the very nice things about SPICE is that both input and output formats are plain-
text, which is the most universal and easy-to-edit electronic format around. Practically any
computer will be able to edit and display this format, even if the SPICE program itself is not
resident on that computer. If the user desires, he or she is free to use the advanced capabilities
of word processing programs to make the output look fancier. Comments can even be inserted
between lines of the output for further clarity to the reader.
75.
7.4. THE COMMAND-LINE INTERFACE 67
7.4 The command-line interface
If you’ve used DOS or UNIX operating systems before in a command-line shell environment,
you may wonder why we have to use the ”<” symbol between the word ”spice” and the name
of the netlist ﬁle to be interpreted. Why not just enter the ﬁle name as the ﬁrst argument
to the command ”spice” as we do when we invoke the text editor? The answer is that SPICE
has the option of an interactive mode, whereby each line of the netlist can be interpreted as
it is entered through the computer’s Standard Input (stdin). If you simple type ”spice” at the
prompt and press [Enter], SPICE will begin to interpret anything you type in to it (live).
For most applications, its nice to save your netlist work in a separate ﬁle and then let SPICE
interpret that ﬁle when you’re ready. This is the way I encourage SPICE to be used, and so
this is the way its presented in this lesson. In order to use SPICE this way in a command-line
environment, we need to use the ”<” redirection symbol to direct the contents of your netlist
ﬁle to Standard Input (stdin), which SPICE can then process.
7.5 Circuit components
Remember that this tutorial is not exhaustive by any means, and that all descriptions for
elements in the SPICE language are documented here in condensed form. SPICE is a very
capable piece of software with lots of options, and I’m only going to document a few of them.
All components in a SPICE source ﬁle are primarily identiﬁed by the ﬁrst letter in each
respective line. Characters following the identifying letter are used to distinguish one compo-
nent of a certain type from another of the same type (r1, r2, r3, rload, rpullup, etc.), and need
not follow any particular naming convention, so long as no more than eight characters are used
in both the component identifying letter and the distinguishing name.
For example, suppose you were simulating a digital circuit with ”pullup” and ”pulldown”
resistors. The name rpullup would be valid because it is seven characters long. The name
rpulldown, however, is nine characters long. This may cause problems when SPICE inter-
prets the netlist.
You can actually get away with component names in excess of eight total characters if there
are no other similarly-named components in the source ﬁle. SPICE only pays attention to the
ﬁrst eight characters of the ﬁrst ﬁeld in each line, so rpulldown is actually interpreted as
rpulldow with the ”n” at the end being ignored. Therefore, any other resistor having the
ﬁrst eight characters in its ﬁrst ﬁeld will be seen by SPICE as the same resistor, deﬁned twice,
which will cause an error (i.e. rpulldown1 and rpulldown2 would be interpreted as the same
name, rpulldow).
It should also be noted that SPICE ignores character case, so r1 and R1 are interpreted by
SPICE as one and the same.
SPICE allows the use of metric preﬁxes in specifying component values, which is a very
handy feature. However, the preﬁx convention used by SPICE differs somewhat from stan-
dard metric symbols, primarily due to the fact that netlists are restricted to standard ASCII
characters (ruling out Greek letters such as µ for the preﬁx ”micro”) and that SPICE is case-
insensitive, so ”m” (which is the standard symbol for ”milli”) and ”M” (which is the standard
symbol for ”Mega”) are interpreted identically. Here are a few examples of preﬁxes used in
SPICE netlists:
76.
68 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
r1 1 0 2t (Resistor R1 , 2t = 2 Tera-ohms = 2 TΩ)
r2 1 0 4g (Resistor R2 , 4g = 4 Giga-ohms = 4 GΩ)
r3 1 0 47meg (Resistor R3 , 47meg = 47 Mega-ohms = 47 MΩ)
r4 1 0 3.3k (Resistor R4 , 3.3k = 3.3 kilo-ohms = 3.3 kΩ)
r5 1 0 55m (Resistor R5 , 55m = 55 milli-ohms = 55 mΩ)
r6 1 0 10u (Resistor R6 , 10u = 10 micro-ohms 10 µΩ)
r7 1 0 30n (Resistor R7 , 30n = 30 nano-ohms = 30 nΩ)
r8 1 0 5p (Resistor R8 , 5p = 5 pico-ohms = 5 pΩ)
r9 1 0 250f (Resistor R9 , 250f = 250 femto-ohms = 250 fΩ)
Scientiﬁc notation is also allowed in specifying component values. For example:
r10 1 0 4.7e3 (Resistor R10 , 4.7e3 = 4.7 x 103 ohms = 4.7 kilo-ohms = 4.7 kΩ)
r11 1 0 1e-12 (Resistor R11 , 1e-12 = 1 x 10−12 ohms = 1 pico-ohm = 1 pΩ)
The unit (ohms, volts, farads, henrys, etc.) is automatically determined by the type of
component being speciﬁed. SPICE ”knows” that all of the above examples are ”ohms” because
they are all resistors (r1, r2, r3, . . . ). If they were capacitors, the values would be interpreted
as ”farads,” if inductors, then ”henrys,” etc.
7.5.1 Passive components
CAPACITORS
General form: c[name] [node1] [node2] [value] ic=[initial voltage]
Example 1: c1 12 33 10u
Example 2: c1 12 33 10u ic=3.5
Comments: The ”initial condition” (ic=) variable is the capacitor’s voltage in units of volts at
the start of DC analysis. It is an optional value, with the starting voltage assumed to be zero if
unspeciﬁed. Starting current values for capacitors are interpreted by SPICE only if the .tran
analysis option is invoked (with the ”uic” option).
INDUCTORS
General form: l[name] [node1] [node2] [value] ic=[initial current]
Example 1: l1 12 33 133m
Example 2: l1 12 33 133m ic=12.7m
Comments: The ”initial condition” (ic=) variable is the inductor’s current in units of amps at
the start of DC analysis. It is an optional value, with the starting current assumed to be zero
if unspeciﬁed. Starting current values for inductors are interpreted by SPICE only if the .tran
analysis option is invoked.
77.
7.5. CIRCUIT COMPONENTS 69
INDUCTOR COUPLING (transformers)
General form: k[name] l[name] l[name] [coupling factor]
Example 1: k1 l1 l2 0.999
Comments: SPICE will only allow coupling factor values between 0 and 1 (non-inclusive),
with 0 representing no coupling and 1 representing perfect coupling. The order of specifying
coupled inductors (l1, l2 or l2, l1) is irrelevant.
RESISTORS
General form: r[name] [node1] [node2] [value]
Example: rload 23 15 3.3k
Comments: In case you were wondering, there is no declaration of resistor power dissipation
rating in SPICE. All components are assumed to be indestructible. If only real life were this
forgiving!
7.5.2 Active components
All semiconductor components must have their electrical characteristics described in a line
starting with the word ”.model”, which tells SPICE exactly how the device will behave. What-
ever parameters are not explicitly deﬁned in the .model card will default to values pre-
programmed in SPICE. However, the .model card must be included, and at least specify the
model name and device type (d, npn, pnp, njf, pjf, nmos, or pmos).
DIODES
General form: d[name] [anode] [cathode] [model]
Example: d1 1 2 mod1
DIODE MODELS:
General form: .model [modelname] d [parmtr1=x] [parmtr2=x] . . .
Example: .model mod1 d
Example: .model mod2 d vj=0.65 rs=1.3
¡hypertarget¿diodeparameter¡/hypertarget¿
Parameter deﬁnitions:
is = saturation current in amps
rs = junction resistance in ohms
n = emission coefﬁcient (unitless)
tt = transit time in seconds
cjo = zero-bias junction capacitance in farads
vj = junction potential in volts
m = grading coefﬁcient (unitless)
eg = activation energy in electron-volts
78.
70 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
xti = saturation-current temperature exponent (unitless)
kf = ﬂicker noise coefﬁcient (unitless)
af = ﬂicker noise exponent (unitless)
fc = forward-bias depletion capacitance coefﬁcient (unitless)
bv = reverse breakdown voltage in volts
ibv = current at breakdown voltage in amps
Comments: The model name must begin with a letter, not a number. If you plan to specify
a model for a 1N4003 rectifying diode, for instance, you cannot use ”1n4003” for the model
name. An alternative might be ”m1n4003” instead.
TRANSISTORS, bipolar junction – BJT
General form: q[name] [collector] [base] [emitter] [model]
Example: q1 2 3 0 mod1
BJT TRANSISTOR MODELS:
General form: .model [modelname] [npn or pnp] [parmtr1=x] . . .
Example: .model mod1 pnp
Example: .model mod2 npn bf=75 is=1e-14
The model examples shown above are very nonspeciﬁc. To accurately model real-life tran-
sistors, more parameters are necessary. Take these two examples, for the popular 2N2222 and
2N2907 transistors (the ”+”) characters represent line-continuation marks in SPICE, when you
wish to break a single line (card) into two or more separate lines on your text editor:
Example: .model m2n2222 npn is=19f bf=150 vaf=100 ikf=.18
+ ise=50p ne=2.5 br=7.5 var=6.4 ikr=12m
+ isc=8.7p nc=1.2 rb=50 re=0.4 rc=0.4 cje=26p
+ tf=0.5n cjc=11p tr=7n xtb=1.5 kf=0.032f af=1
Example: .model m2n2907 pnp is=1.1p bf=200 nf=1.2 vaf=50
+ ikf=0.1 ise=13p ne=1.9 br=6 rc=0.6 cje=23p
+ vje=0.85 mje=1.25 tf=0.5n cjc=19p vjc=0.5
+ mjc=0.2 tr=34n xtb=1.5
Parameter deﬁnitions:
is = transport saturation current in amps
bf = ideal maximum forward Beta (unitless)
nf = forward current emission coefﬁcient (unitless)
vaf = forward Early voltage in volts
ikf = corner for forward Beta high-current rolloff in amps
ise = B-E leakage saturation current in amps
ne = B-E leakage emission coefﬁcient (unitless)
br = ideal maximum reverse Beta (unitless)
nr = reverse current emission coefﬁcient (unitless)
79.
7.5. CIRCUIT COMPONENTS 71
bar = reverse Early voltage in volts
ikrikr = corner for reverse Beta high-current rolloff in amps
iscisc = B-C leakage saturation current in amps
nc = B-C leakage emission coefﬁcient (unitless)
rb = zero bias base resistance in ohms
irb = current for base resistance halfway value in amps
rbm = minimum base resistance at high currents in ohms
re = emitter resistance in ohms
rc = collector resistance in ohms
cje = B-E zero-bias depletion capacitance in farads
vje = B-E built-in potential in volts
mje = B-E junction exponential factor (unitless)
tf = ideal forward transit time (seconds)
xtf = coefﬁcient for bias dependence of transit time (unitless)
vtf = B-C voltage dependence on transit time, in volts
itf = high-current parameter effect on transit time, in amps
ptf = excess phase at f=1/(transit time)(2)(pi) Hz, in degrees
cjc = B-C zero-bias depletion capacitance in farads
vjc = B-C built-in potential in volts
mjc = B-C junction exponential factor (unitless)
xjcj = B-C depletion capacitance fraction connected in base node (unitless)
tr = ideal reverse transit time in seconds
cjs = zero-bias collector-substrate capacitance in farads
vjs = substrate junction built-in potential in volts
mjs = substrate junction exponential factor (unitless)
xtb = forward/reverse Beta temperature exponent
eg = energy gap for temperature effect on transport saturation current in electron-volts
xti = temperature exponent for effect on transport saturation current (unitless)
kf = ﬂicker noise coefﬁcient (unitless)
af = ﬂicker noise exponent (unitless)
fc = forward-bias depletion capacitance formula coefﬁcient (unitless)
Comments: Just as with diodes, the model name given for a particular transistor type
must begin with a letter, not a number. That’s why the examples given above for the 2N2222
and 2N2907 types of BJTs are named ”m2n2222” and ”q2n2907” respectively.
As you can see, SPICE allows for very detailed speciﬁcation of transistor properties. Many
of the properties listed above are well beyond the scope and interest of the beginning electronics
student, and aren’t even useful apart from knowing the equations SPICE uses to model BJT
transistors. For those interested in learning more about transistor modeling in SPICE, consult
other books, such as Andrei Vladimirescu’s The Spice Book (ISBN 0-471-60926-9).
JFET, junction ﬁeld-effect transistor
General form: j[name] [drain] [gate] [source] [model]
Example: j1 2 3 0 mod1
80.
72 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
JFET TRANSISTOR MODELS:
General form: .model [modelname] [njf or pjf] [parmtr1=x] . . .
Example: .model mod1 pjf
Example: .model mod2 njf lambda=1e-5 pb=0.75
Parameter deﬁnitions:
vto = threshold voltage in volts
beta = transconductance parameter in amps/volts2
lambda = channel length modulation parameter in units of 1/volts
rd = drain resistance in ohms
rs = source resistance in ohms
cgs = zero-bias G-S junction capacitance in farads
cgd = zero-bias G-D junction capacitance in farads
pb = gate junction potential in volts
is = gate junction saturation current in amps
kf = ﬂicker noise coefﬁcient (unitless)
af = ﬂicker noise exponent (unitless)
fc = forward-bias depletion capacitance coefﬁcient (unitless)
MOSFET, transistor
General form: m[name] [drain] [gate] [source] [substrate] [model]
Example: m1 2 3 0 0 mod1
MOSFET TRANSISTOR MODELS:
General form: .model [modelname] [nmos or pmos] [parmtr1=x] . . .
Example: .model mod1 pmos
Example: .model mod2 nmos level=2 phi=0.65 rd=1.5
Example: .model mod3 nmos vto=-1 (depletion)
Example: .model mod4 nmos vto=1 (enhancement)
Example: .model mod5 pmos vto=1 (depletion)
Example: .model mod6 pmos vto=-1 (enhancement)
Comments: In order to distinguish between enhancement mode and depletion-mode (also
known as depletion-enhancement mode) transistors, the model parameter ”vto” (zero-bias
threshold voltage) must be speciﬁed. Its default value is zero, but a positive value (+1 volts,
for example) on a P-channel transistor or a negative value (-1 volts) on an N-channel transis-
tor will specify that transistor to be a depletion (otherwise known as depletion-enhancement)
mode device. Conversely, a negative value on a P-channel transistor or a positive value on an
N-channel transistor will specify that transistor to be an enhancement mode device.
Remember that enhancement mode transistors are normally-off devices, and must be turned
on by the application of gate voltage. Depletion-mode transistors are normally ”on,” but can
be ”pinched off ” as well as enhanced to greater levels of drain current by applied gate voltage,
hence the alternate designation of ”depletion-enhancement” MOSFETs. The ”vto” parameter
speciﬁes the threshold gate voltage for MOSFET conduction.
81.
7.5. CIRCUIT COMPONENTS 73
7.5.3 Sources
AC SINEWAVE VOLTAGE SOURCES (when using .ac card to specify frequency):
General form: v[name] [+node] [-node] ac [voltage] [phase] sin
Example 1: v1 1 0 ac 12 sin
Example 2: v1 1 0 ac 12 240 sin (12 V 240o )
Comments: This method of specifying AC voltage sources works well if you’re using multi-
ple sources at different phase angles from each other, but all at the same frequency. If you need
to specify sources at different frequencies in the same circuit, you must use the next method!
AC SINEWAVE VOLTAGE SOURCES (when NOT using .ac card to specify fre-
quency):
General form: v[name] [+node] [-node] sin([offset] [voltage]
+ [freq] [delay] [damping factor])
Example 1: v1 1 0 sin(0 12 60 0 0)
Parameter deﬁnitions:
offset = DC bias voltage, offsetting the AC waveform by a speciﬁed voltage.
voltage = peak, or crest, AC voltage value for the waveform.
freq = frequency in Hertz.
delay = time delay, or phase offset for the waveform, in seconds.
damping factor = a ﬁgure used to create waveforms of decaying amplitude.
Comments: This method of specifying AC voltage sources works well if you’re using multi-
ple sources at different frequencies from each other. Representing phase shift is tricky, though,
necessitating the use of the delay factor.
DC VOLTAGE SOURCES (when using .dc card to specify voltage):
General form: v[name] [+node] [-node] dc
Example 1: v1 1 0 dc
Comments: If you wish to have SPICE output voltages not in reference to node 0, you must
use the .dc analysis option, and to use this option you must specify at least one of your DC
sources in this manner.
DC VOLTAGE SOURCES (when NOT using .dc card to specify voltage):
General form: v[name] [+node] [-node] dc [voltage]
Example 1: v1 1 0 dc 12
Comments: Nothing noteworthy here!
PULSE VOLTAGE SOURCES
General form: v[name] [+node] [-node] pulse ([i] [p] [td] [tr]
+ [tf] [pw] [pd])
Parameter deﬁnitions:
i = initial value
p = pulse value
td = delay time (all time parameters in units of seconds)
tr = rise time
tf = fall time
82.
74 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
pw = pulse width
pd = period
Example 1: v1 1 0 pulse (-3 3 0 0 0 10m 20m)
Comments: Example 1 is a perfect square wave oscillating between -3 and +3 volts, with
zero rise and fall times, a 20 millisecond period, and a 50 percent duty cycle (+3 volts for 10
ms, then -3 volts for 10 ms).
AC SINEWAVE CURRENT SOURCES (when using .ac card to specify frequency):
General form: i[name] [+node] [-node] ac [current] [phase] sin
Example 1: i1 1 0 ac 3 sin (3 amps)
Example 2: i1 1 0 ac 1m 240 sin (1 mA 240o )
Comments: The same comments apply here (and in the next example) as for AC voltage
sources.
AC SINEWAVE CURRENT SOURCES (when NOT using .ac card to specify fre-
quency):
General form: i[name] [+node] [-node] sin([offset]
+ [current] [freq] 0 0)
Example 1: i1 1 0 sin(0 1.5 60 0 0)
DC CURRENT SOURCES (when using .dc card to specify current):
General form: i[name] [+node] [-node] dc
Example 1: i1 1 0 dc
DC CURRENT SOURCES (when NOT using .dc card to specify current):
General form: i[name] [+node] [-node] dc [current]
Example 1: i1 1 0 dc 12
Comments: Even though the books all say that the ﬁrst node given for the DC current
source is the positive node, that’s not what I’ve found to be in practice. In actuality, a DC
current source in SPICE pushes current in the same direction as a voltage source (battery)
would with its negative node speciﬁed ﬁrst.
PULSE CURRENT SOURCES
General form: i[name] [+node] [-node] pulse ([i] [p] [td] [tr]
+ [tf] [pw] [pd])
Parameter deﬁnitions:
i = initial value
p = pulse value
td = delay time
tr = rise time
tf = fall time
pw = pulse width
pd = period
Example 1: i1 1 0 pulse (-3m 3m 0 0 0 17m 34m)
83.
7.6. ANALYSIS OPTIONS 75
Comments: Example 1 is a perfect square wave oscillating between -3 mA and +3 mA,
with zero rise and fall times, a 34 millisecond period, and a 50 percent duty cycle (+3 mA for
17 ms, then -3 mA for 17 ms).
VOLTAGE SOURCES (dependent):
General form: e[name] [out+node] [out-node] [in+node] [in-node]
+ [gain]
Example 1: e1 2 0 1 2 999k
Comments: Dependent voltage sources are great to use for simulating operational ampli-
ﬁers. Example 1 shows how such a source would be conﬁgured for use as a voltage follower,
inverting input connected to output (node 2) for negative feedback, and the noninverting input
coming in on node 1. The gain has been set to an arbitrarily high value of 999,000. One word
of caution, though: SPICE does not recognize the input of a dependent source as being a load,
so a voltage source tied only to the input of an independent voltage source will be interpreted
as ”open.” See op-amp circuit examples for more details on this.
CURRENT SOURCES (dependent):
7.6 Analysis options
AC ANALYSIS:
General form: .ac [curve] [points] [start] [final]
Example 1: .ac lin 1 1000 1000
Comments: The [curve] ﬁeld can be ”lin” (linear), ”dec” (decade), or ”oct” (octave), specify-
ing the (non)linearity of the frequency sweep. ¡points¿ speciﬁes how many points within the
frequency sweep to perform analyses at (for decade sweep, the number of points per decade;
for octave, the number of points per octave). The [start] and [ﬁnal] ﬁelds specify the starting
and ending frequencies of the sweep, respectively. One ﬁnal note: the ”start” value cannot be
zero!
DC ANALYSIS:
General form: .dc [source] [start] [final] [increment]
Example 1: .dc vin 1.5 15 0.5
Comments: The .dc card is necessary if you want to print or plot any voltage between
two nonzero nodes. Otherwise, the default ”small-signal” analysis only prints out the voltage
between each nonzero node and node zero.
TRANSIENT ANALYSIS:
General form: .tran [increment] [stop time] [start time]
+ [comp interval]
Example 1: .tran 1m 50m uic
Example 2: .tran .5m 32m 0 .01m
Comments: Example 1 has an increment time of 1 millisecond and a stop time of 50 mil-
liseconds (when only two parameters are speciﬁed, they are increment time and stop time,
84.
76 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
respectively). Example 2 has an increment time of 0.5 milliseconds, a stop time of 32 mil-
liseconds, a start time of 0 milliseconds (no delay on start), and a computation interval of 0.01
milliseconds.
Default value for start time is zero. Transient analysis always beings at time zero, but
storage of data only takes place between start time and stop time. Data output interval is
increment time, or (stop time - start time)/50, which ever is smallest. However, the computing
interval variable can be used to force a computational interval smaller than either. For large
total interval counts, the itl5 variable in the .options card may be set to a higher number.
The ”uic” option tells SPICE to ”use initial conditions.”
PLOT OUTPUT:
General form: .plot [type] [output1] [output2] . . . [output n]
Example 1: .plot dc v(1,2) i(v2)
Example 2: .plot ac v(3,4) vp(3,4) i(v1) ip(v1)
Example 3: .plot tran v(4,5) i(v2)
Comments: SPICE can’t handle more than eight data point requests on a single .plot or
.print card. If requesting more than eight data points, use multiple cards!
Also, here’s a major caveat when using SPICE version 3: if you’re performing AC analysis
and you ask SPICE to plot an AC voltage as in example #2, the v(3,4) command will only
output the real component of a rectangular-form complex number! SPICE version 2 outputs
the polar magnitude of a complex number: a much more meaningful quantity if only a single
quantity is asked for. To coerce SPICE3 to give you polar magnitude, you will have to re-write
the .print or .plot argument as such: vm(3,4).
PRINT OUTPUT:
General form: .print [type] [output1] [output2] . . . [output n]
Example 1: .print dc v(1,2) i(v2)
Example 2: .print ac v(2,4) i(vinput) vp(2,3)
Example 3: .print tran v(4,5) i(v2)
Comments: SPICE can’t handle more than eight data point requests on a single .plot or
.print card. If requesting more than eight data points, use multiple cards!
FOURIER ANALYSIS:
General form: .four [freq] [output1] [output2] . . . [output n]
Example 1: .four 60 v(1,2)
Comments: The .four card relies on the .tran card being present somewhere in the
deck, with the proper time periods for analysis of adequate cycles. Also, SPICE may ”crash” if
a .plot analysis isn’t done along with the .four analysis, even if all .tran parameters are
technically correct. Finally, the .four analysis option only works when the frequency of the
AC source is speciﬁed in that source’s card line, and not in an .ac analysis option line.
It helps to include a computation interval variable in the .tran card for better analysis
precision. A Fourier analysis of the voltage or current speciﬁed is performed up to the 9th
harmonic, with the [freq] speciﬁcation being the fundamental, or starting frequency of the
analysis spectrum.
MISCELLANEOUS:
85.
7.6. ANALYSIS OPTIONS 77
General form: .options [option1] [option2]
Example 1: .options limpts=500
Example 2: .options itl5=0
Example 3: .options method=gear
Example 4: .options list
Example 5: .options nopage
Example 6: .options numdgt=6
Comments: There are lots of options that can be speciﬁed using this card. Perhaps the one
most needed by beginning users of SPICE is the ”limpts” setting. When running a simulation
that requires more than 201 points to be printed or plotted, this calculation point limit must
be increased or else SPICE will terminate analysis. The example given above (limpts=500)
tells SPICE to allocate enough memory to handle at least 500 calculation points in whatever
type of analysis is speciﬁed (DC, AC, or transient).
In example 2, we see an iteration variable (itl5) being set to a value of 0. There are
actually six different iteration variables available for user manipulation. They control the
iteration cycle limits for solution of nonlinear equations. The variable itl5 sets the maximum
number of iterations for a transient analysis. Similar to the limpts variable, itl5 usually
needs to be set when a small computation interval has been speciﬁed on a .tran card. Setting
itl5 to a value of 0 turns off the limit entirely, allowing the computer inﬁnite iteration cycles
(inﬁnite time) to compute the analysis. Warning: this may result in long simulation times!
Example 3 with ”method=gear” sets the numerical integration method used by SPICE. The
default is ”trapezoid” rather than ”gear,” trapezoid being a simple geometric approximation of
area under a curve found by slicing up the curve into trapezoids to approximate the shape. The
”gear” method is based on second-order or better polynomial equations and is named after C.W.
Gear (Numerical Integration of Stiff Ordinary Equations, Report 221, Department of Computer
Science, University of Illinois, Urbana). The Gear method of integration is more demanding of
the computer (computationally ”expensive”) and will sometimes give slightly different results
from the trapezoid method.
The ”list” option shown in example 4 gives a verbose summary of all circuit components
and their respective values in the ﬁnal output.
By default, SPICE will insert ASCII page-break control codes in the output to separate
different sections of the analysis. Specifying the ”nopage” option (example 5) will prevent
such pagination.
The ”numdgt” option shown in example 6 speciﬁes the number of signiﬁcant digits out-
put when using one of the ”.print” data output options. SPICE defaults at a precision of 4
signiﬁcant digits.
WIDTH CONTROL:
General form: .width in=[columns] out=[columns]
Example 1: .width out=80
Comments: The .width card can be used to control the width of text output lines upon
analysis. This is especially handy when plotting graphs with the .plot card. The default
value is 120, which can cause problems on 80-character terminal displays unless set to 80 with
this command.
86.
78 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
7.7 Quirks
”Garbage in, garbage out.”
Anonymous
SPICE is a very reliable piece of software, but it does have its little quirks that take some
getting used to. By ”quirk” I mean a demand placed upon the user to write the source ﬁle in a
particular way in order for it to work without giving error messages. I do not mean any kind
of fault with SPICE which would produce erroneous or misleading results: that would be more
properly referred to as a ”bug.” Speaking of bugs, SPICE has a few of them as well.
Some (or all) of these quirks may be unique to SPICE version 2g6, which is the only version
I’ve used extensively. They may have been ﬁxed in later versions.
7.7.1 A good beginning
SPICE demands that the source ﬁle begin with something other than the ﬁrst ”card” in the
circuit description ”deck.” This ﬁrst character in the source ﬁle can be a linefeed, title line, or
a comment: there just has to be something there before the ﬁrst component-specifying line of
the ﬁle. If not, SPICE will refuse to do an analysis at all, claiming that there is a serious error
(such as improper node connections) in the ”deck.”
7.7.2 A good ending
SPICE demands that the .end line at the end of the source ﬁle not be terminated with a line-
feed or carriage return character. In other words, when you ﬁnish typing ”.end” you should
not hit the [Enter] key on your keyboard. The cursor on your text editor should stop imme-
diately to the right of the ”d” after the ”.end” and go no further. Failure to heed this quirk
will result in a ”missing .end card” error message at the end of the analysis output. The actual
circuit analysis is not affected by this error, so I normally ignore the message. However, if
you’re looking to receive a ”perfect” output, you must pay heed to this idiosyncrasy.
7.7.3 Must have a node 0
You are given much freedom in numbering circuit nodes, but you must have a node 0 some-
where in your netlist in order for SPICE to work. Node 0 is the default node for circuit ground,
and it is the point of reference for all voltages speciﬁed at single node locations.
When simple DC analysis is performed by SPICE, the output will contain a listing of volt-
ages at all non-zero nodes in the circuit. The point of reference (ground) for all these voltage
readings is node 0. For example:
node voltage node voltage
( 1) 15.0000 ( 2) 0.6522
In this analysis, there is a DC voltage of 15 volts between node 1 and ground (node 0), and
a DC voltage of 0.6522 volts between node 2 and ground (node 0). In both these cases, the
voltage polarity is negative at node 0 with reference to the other node (in other words, both
nodes 1 and 2 are positive with respect to node 0).
87.
7.7. QUIRKS 79
7.7.4 Avoid open circuits
SPICE cannot handle open circuits of any kind. If your netlist speciﬁes a circuit with an open
voltage source, for example, SPICE will refuse to perform an analysis. A prime example of this
type of error is found when ”connecting” a voltage source to the input of a voltage-dependent
source (used to simulate an operational ampliﬁer). SPICE needs to see a complete path for
current, so I usually tie a high-value resistor (call it rbogus!) across the voltage source to act
as a minimal load.
7.7.5 Avoid certain component loops
SPICE cannot handle certain uninterrupted loops of components in a circuit, namely voltage
sources and inductors. The following loops will cause SPICE to abort analysis:
Parallel inductors
2 2 2
L1 10 mH L2 50 mH L3 25 mH
4 4 4
netlist
l1 2 4 10m
l2 2 4 50m
l3 2 4 25m
Voltage source / inductor loop
1 1
V1 12 V L1 150 mH
0 0
netlist
v1 1 0 dc 12
l1 1 0 150m
88.
80 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
Series capacitors
5
C1 33 µF
6
C2 47 µF
7
netlist
c1 5 6 33u
c2 6 7 47u
The reason SPICE can’t handle these conditions stems from the way it performs DC anal-
ysis: by treating all inductors as shorts and all capacitors as opens. Since short-circuits (0 Ω)
and open circuits (inﬁnite resistance) either contain or generate mathematical inﬁnitudes, a
computer simply cannot deal with them, and so SPICE will discontinue analysis if any of these
conditions occur.
In order to make these component conﬁgurations acceptable to SPICE, you must insert
resistors of appropriate values into the appropriate places, eliminating the respective short-
circuits and open-circuits. If a series resistor is required, choose a very low resistance value.
Conversely, if a parallel resistor is required, choose a very high resistance value. For example:
To ﬁx the parallel inductor problem, insert a very low-value resistor in series with each
offending inductor.
89.
7.7. QUIRKS 81
Original circuit
2 2 2
L1 10 mH L2 50 mH L3 25 mH
4 4 4
"Fixed" circuit
3 Rbogus1 2 Rbogus2 5
L1 10 mH L2 50 mH L3 25 mH
4 4 4
original netlist
l1 2 4 10m
l2 2 4 50m
l3 2 4 25m
fixed netlist
rbogus1 2 3 1e-12
rbogus2 2 5 1e-12
l1 3 4 10m
l2 2 4 50m
l3 5 4 25m
The extremely low-resistance resistors Rbogus1 and Rbogus2 (each one with a mere 1 pico-ohm
of resistance) ”break up” the direct parallel connections that existed between L1 , L2 , and L3 . It
is important to choose very low resistances here so that circuit operation is not substantially
impacted by the ”ﬁx.”
To ﬁx the voltage source / inductor loop, insert a very low-value resistor in series with the
two components.
90.
82 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
Original circuit
1 1
V1 12 V L1 150 mH
0 0
"Fixed" circuit
Rbogus
1 2
V1 12 V L1 150 mH
0 0
original netlist
v1 1 0 dc 12
l1 1 0 150m
fixed netlist
v1 1 0 dc 12
l1 2 0 150m
rbogus 1 2 1e-12
As in the previous example with parallel inductors, it is important to make the correction
resistor (Rbogus ) very low in resistance, so as to not substantially impact circuit operation.
To ﬁx the series capacitor circuit, one of the capacitors must have a resistor shunting across
it. SPICE requires a DC current path to each capacitor for analysis.
91.
7.7. QUIRKS 83
Original circuit "Fixed" circuit
5 5
C1 33 µF C1 33 µF
6 6
6
C2 47 µF C2 47 µF Rbogus
7
7 7
original netlist
c1 5 6 33u
c2 6 7 47u
fixed netlist
c1 5 6 33u
c2 6 7 47u
rbogus 6 7 9e12
The Rbogus value of 9 Tera-ohms provides a DC current path to C1 (and around C2 ) without
substantially impacting the circuit’s operation.
7.7.6 Current measurement
Although printing or plotting of voltage is quite easy in SPICE, the output of current values is
a bit more difﬁcult. Voltage measurements are speciﬁed by declaring the appropriate circuit
nodes. For example, if we desire to know the voltage across a capacitor whose leads connect
between nodes 4 and 7, we might make out .print statement look like this:
C1
4 7
22 µF
c1 4 7 22u
.print ac v(4,7)
However, if we wanted to have SPICE measure the current through that capacitor, it wouldn’t
be quite so easy. Currents in SPICE must be speciﬁed in relation to a voltage source, not any
arbitrary component. For example:
92.
84 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
Vinput C1
6 4 7
I 22 µF
c1 4 7 22u
vinput 6 4 ac 1 sin
.print ac i(vinput)
This .print card instructs SPICE to print the current through voltage source Vinput , which
happens to be the same as the current through our capacitor between nodes 4 and 7. But what
if there is no such voltage source in our circuit to reference for current measurement? One
solution is to insert a shunt resistor into the circuit and measure voltage across it. In this case,
I have chosen a shunt resistance value of 1 Ω to produce 1 volt per amp of current through C1 :
Rshunt C1
6 4 7
1Ω 22 µF
I
c1 4 7 22u
rshunt 6 4 1
.print ac v(6,4)
However, the insertion of an extra resistance into our circuit large enough to drop a mean-
ingful voltage for the intended range of current might adversely affect things. A better solution
for SPICE is this, although one would never seek such a current measurement solution in real
life:
Vbogus C1
6 4 7
0V 22 µF
I
c1 4 7 22u
vbogus 6 4 dc 0
.print ac i(vbogus)
Inserting a ”bogus” DC voltage source of zero volts doesn’t affect circuit operation at all, yet
it provides a convenient place for SPICE to take a current measurement. Interestingly enough,
it doesn’t matter that Vbogus is a DC source when we’re looking to measure AC current! The
fact that SPICE will output an AC current reading is determined by the ”ac” speciﬁcation in
the .print card and nothing more.
It should also be noted that the way SPICE assigns a polarity to current measurements is
a bit odd. Take the following circuit as an example:
93.
7.7. QUIRKS 85
R1
1 2
5 kΩ
V1 10 V R2 5 kΩ
0 0
example
v1 1 0
r1 1 2 5k
r2 2 0 5k
.dc v1 10 10 1
.print dc i(v1)
.end
With 10 volts total voltage and 10 kΩ total resistance, you might expect SPICE to tell you
there’s going to be 1 mA (1e-03) of current through voltage source V1 , but in actuality SPICE
will output a ﬁgure of negative 1 mA (-1e-03)! SPICE regards current out of the negative end
of a DC voltage source (the normal direction) to be a negative value of current rather than a
positive value of current. There are times I’ll throw in a ”bogus” voltage source in a DC circuit
like this simply to get SPICE to output a positive current value:
R1
1 2
5 kΩ
V1 10 V R2 5 kΩ
Vbogus
0 3
0V
example
v1 1 0
r1 1 2 5k
r2 2 3 5k
vbogus 3 0 dc 0
.dc v1 10 10 1
.print dc i(vbogus)
.end
Notice how Vbogus is positioned so that the circuit current will enter its positive side (node
3) and exit its negative side (node 0). This orientation will ensure a positive output ﬁgure for
circuit current.
94.
86 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
7.7.7 Fourier analysis
When performing a Fourier (frequency-domain) analysis on a waveform, I have found it neces-
sary to either print or plot the waveform using the .print or .plot cards, respectively. If you
don’t print or plot it, SPICE will pause for a moment during analysis and then abort the job
after outputting the ”initial transient solution.”
Also, when analyzing a square wave produced by the ”pulse” source function, you must
give the waveform some ﬁnite rise and fall time, or else the Fourier analysis results will be
incorrect. For some reason, a perfect square wave with zero rise/fall time produces signiﬁcant
levels of even harmonics according to SPICE’s Fourier analysis option, which is not true for
real square waves.
7.8 Example circuits and netlists
The following circuits are pre-tested netlists for SPICE 2g6, complete with short descriptions
when necessary. Feel free to ”copy” and ”paste” any of the netlists to your own SPICE source ﬁle
for analysis and/or modiﬁcation. My goal here is twofold: to give practical examples of SPICE
netlist design to further understanding of SPICE netlist syntax, and to show how simple and
compact SPICE netlists can be in analyzing simple circuits.
All output listings for these examples have been ”trimmed” of extraneous information, giv-
ing you the most succinct presentation of the SPICE output as possible. I do this primarily
to save space on this document. Typical SPICE outputs contain lots of headers and summary
information not necessarily germane to the task at hand. So don’t be surprised when you run
a simulation on your own and ﬁnd that the output doesn’t exactly look like what I have shown
here!
7.8.1 Multiple-source DC resistor network, part 1
R1 2 R2
1 3
10 kΩ 8.1 kΩ
V1 24 V R3 4.7 kΩ V2 15 V
0 0 0
Without a .dc card and a .print or .plot card, the output for this netlist will only display
voltages for nodes 1, 2, and 3 (with reference to node 0, of course).
Netlist:
Multiple dc sources
v1 1 0 dc 24
v2 3 0 dc 15
95.
7.8. EXAMPLE CIRCUITS AND NETLISTS 87
r1 1 2 10k
r2 2 3 8.1k
r3 2 0 4.7k
.end
Output:
node voltage node voltage node voltage
( 1) 24.0000 ( 2) 9.7470 ( 3) 15.0000
voltage source currents
name current
v1 -1.425E-03
v2 -6.485E-04
total power dissipation 4.39E-02 watts
7.8.2 Multiple-source DC resistor network, part 2
R1 2 R2
1 3
10 kΩ 8.1 kΩ
V1 24 V R3 4.7 kΩ V2 15 V
0 0 0
By adding a .dc analysis card and specifying source V1 from 24 volts to 24 volts in 1 step
(in other words, 24 volts steady), we can use the .print card analysis to print out voltages
between any two points we desire. Oddly enough, when the .dc analysis option is invoked,
the default voltage printouts for each node (to ground) disappears, so we end up having to
explicitly specify them in the .print card to see them at all.
Netlist:
Multiple dc sources
v1 1 0
v2 3 0 15
r1 1 2 10k
r2 2 3 8.1k
r3 2 0 4.7k
.dc v1 24 24 1
.print dc v(1) v(2) v(3) v(1,2) v(2,3)
.end
Output:
96.
88 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
v1 v(1) v(2) v(3) v(1,2) v(2,3)
2.400E+01 2.400E+01 9.747E+00 1.500E+01 1.425E+01 -5.253E+00
7.8.3 RC time-constant circuit
1 1 1
V1 10 V C1 C2
47 µF 22 µF
R1
0 2 2
3.3 kΩ
For DC analysis, the initial conditions of any reactive component (C or L) must be speciﬁed
(voltage for capacitors, current for inductors). This is provided by the last data ﬁeld of each
capacitor card (ic=0). To perform a DC analysis, the .tran (”transient”) analysis option must
be speciﬁed, with the ﬁrst data ﬁeld specifying time increment in seconds, the second specifying
total analysis timespan in seconds, and the ”uic” telling it to ”use initial conditions” when
analyzing.
Netlist:
RC time delay circuit
v1 1 0 dc 10
c1 1 2 47u ic=0
c2 1 2 22u ic=0
r1 2 0 3.3k
.tran .05 1 uic
.print tran v(1,2)
.end
Output:
time v(1,2)
0.000E+00 7.701E-06
5.000E-02 1.967E+00
1.000E-01 3.551E+00
1.500E-01 4.824E+00
2.000E-01 5.844E+00
2.500E-01 6.664E+00
3.000E-01 7.322E+00
3.500E-01 7.851E+00
4.000E-01 8.274E+00
4.500E-01 8.615E+00
5.000E-01 8.888E+00
5.500E-01 9.107E+00
97.
7.8. EXAMPLE CIRCUITS AND NETLISTS 89
6.000E-01 9.283E+00
6.500E-01 9.425E+00
7.000E-01 9.538E+00
7.500E-01 9.629E+00
8.000E-01 9.702E+00
8.500E-01 9.761E+00
9.000E-01 9.808E+00
9.500E-01 9.846E+00
1.000E+00 9.877E+00
7.8.4 Plotting and analyzing a simple AC sinewave voltage
1 1
V1
15 V Rload 10 kΩ
60 Hz
0 0
This exercise does show the proper setup for plotting instantaneous values of a sine-wave
voltage source with the .plot function (as a transient analysis). Not surprisingly, the Fourier
analysis in this deck also requires the .tran (transient) analysis option to be speciﬁed over
a suitable time range. The time range in this particular deck allows for a Fourier analysis
with rather poor accuracy. The more cycles of the fundamental frequency that the transient
analysis is performed over, the more precise the Fourier analysis will be. This is not a quirk of
SPICE, but rather a basic principle of waveforms.
Netlist:
v1 1 0 sin(0 15 60 0 0)
rload 1 0 10k
* change tran card to the following for better Fourier precision
* .tran 1m 30m .01m and include .options card:
* .options itl5=30000
.tran 1m 30m
.plot tran v(1)
.four 60 v(1)
.end
Output:
time v(1) -2.000E+01 -1.000E+01 0.000E+00 1.000E+01
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0.000E+00 0.000E+00 . . * . .
1.000E-03 5.487E+00 . . . * . .
2.000E-03 1.025E+01 . . . * .
3.000E-03 1.350E+01 . . . . * .
4.000E-03 1.488E+01 . . . . *.
99.
7.8. EXAMPLE CIRCUITS AND NETLISTS 91
7.8.5 Simple AC resistor-capacitor circuit
R1
1 2
30 Ω
V1 12 V C1 100 µF
60 Hz
0 0
The .ac card speciﬁes the points of ac analysis from 60Hz to 60Hz, at a single point. This card,
of course, is a bit more useful for multi-frequency analysis, where a range of frequencies can
be analyzed in steps. The .print card outputs the AC voltage between nodes 1 and 2, and the
AC voltage between node 2 and ground.
Netlist:
Demo of a simple AC circuit
v1 1 0 ac 12 sin
r1 1 2 30
c1 2 0 100u
.ac lin 1 60 60
.print ac v(1,2) v(2)
.end
Output:
freq v(1,2) v(2)
6.000E+01 8.990E+00 7.949E+00
7.8.6 Low-pass ﬁlter
L1 L2
2 3 4
100 mH 250 mH
V1 24 V
1 C1 100 µF Rload 1 kΩ
V2 24 V
0 0 0
100.
92 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
This low-pass ﬁlter blocks AC and passes DC to the Rload resistor. Typical of a ﬁlter used to
suppress ripple from a rectiﬁer circuit, it actually has a resonant frequency, technically making
it a band-pass ﬁlter. However, it works well anyway to pass DC and block the high-frequency
harmonics generated by the AC-to-DC rectiﬁcation process. Its performance is measured with
an AC source sweeping from 500 Hz to 15 kHz. If desired, the .print card can be substituted
or supplemented with a .plot card to show AC voltage at node 4 graphically.
Netlist:
Lowpass filter
v1 2 1 ac 24 sin
v2 1 0 dc 24
rload 4 0 1k
l1 2 3 100m
l2 3 4 250m
c1 3 0 100u
.ac lin 30 500 15k
.print ac v(4)
.plot ac v(4)
.end
freq v(4)
5.000E+02 1.935E-01
1.000E+03 3.275E-02
1.500E+03 1.057E-02
2.000E+03 4.614E-03
2.500E+03 2.402E-03
3.000E+03 1.403E-03
3.500E+03 8.884E-04
4.000E+03 5.973E-04
4.500E+03 4.206E-04
5.000E+03 3.072E-04
5.500E+03 2.311E-04
6.000E+03 1.782E-04
6.500E+03 1.403E-04
7.000E+03 1.124E-04
7.500E+03 9.141E-05
8.000E+03 7.536E-05
8.500E+03 6.285E-05
9.000E+03 5.296E-05
9.500E+03 4.504E-05
1.000E+04 3.863E-05
1.050E+04 3.337E-05
1.100E+04 2.903E-05
1.150E+04 2.541E-05
1.200E+04 2.237E-05
1.250E+04 1.979E-05
102.
94 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
7.8.7 Multiple-source AC network
L1 L2
1 2 3
450 mH 150 mH
+ +
V1 55 V C1 330 µF V2 43 V
30 Hz 30 Hz
- 0o - 25o
0 0 0
One of the idiosyncrasies of SPICE is its inability to handle any loop in a circuit exclusively
composed of series voltage sources and inductors. Therefore, the ”loop” of V1 -L1 -L2 -V2 -V1 is
unacceptable. To get around this, I had to insert a low-resistance resistor somewhere in that
loop to break it up. Thus, we have Rbogus between 3 and 4 (with 1 pico-ohm of resistance), and
V2 between 4 and 0. The circuit above is the original design, while the circuit below has Rbogus
inserted to avoid the SPICE error.
L1 L2
1 2 3
450 mH 150 mH
Rbogus 1 pΩ
+
V1 55 V C1 330 µF 4
30 Hz +
- 0o V2 43 V
30 Hz
- 25o
0 0 0
Netlist:
Multiple ac source
v1 1 0 ac 55 0 sin
v2 4 0 ac 43 25 sin
l1 1 2 450m
c1 2 0 330u
l2 2 3 150m
103.
7.8. EXAMPLE CIRCUITS AND NETLISTS 95
rbogus 3 4 1e-12
.ac lin 1 30 30
.print ac v(2)
.end
Output:
freq v(2)
3.000E+01 1.413E+02
7.8.8 AC phase shift demonstration
1 1 1
Rshunt1 1 Ω Rshunt2 1Ω
2 3
L1 1H R1 6.3 kΩ
0 0 0
The currents through each leg are indicated by the voltage drops across each respective shunt
resistor (1 amp = 1 volt through 1 Ω), output by the v(1,2) and v(1,3) terms of the .print
card. The phase of the currents through each leg are indicated by the phase of the voltage
drops across each respective shunt resistor, output by the vp(1,2) and vp(1,3) terms in the
.print card.
Netlist:
phase shift
v1 1 0 ac 4 sin
rshunt1 1 2 1
rshunt2 1 3 1
l1 2 0 1
r1 3 0 6.3k
.ac lin 1 1000 1000
.print ac v(1,2) v(1,3) vp(1,2) vp(1,3)
.end
Output:
freq v(1,2) v(1,3) vp(1,2) vp(1,3)
1.000E+03 6.366E-04 6.349E-04 -9.000E+01 0.000E+00
104.
96 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
7.8.9 Transformer circuit
2
Rbogus0 L2 R1
1 1H
L1 4
V1 3
100 H L3
0 R2
25 H
5
Rbogus1 Rbogus2
0 0
SPICE understands transformers as a set of mutually coupled inductors. Thus, to simulate
a transformer in SPICE, you must specify the primary and secondary windings as separate
inductors, then instruct SPICE to link them together with a ”k” card specifying the coupling
constant. For ideal transformer simulation, the coupling constant would be unity (1). However,
SPICE can’t handle this value, so we use something like 0.999 as the coupling factor.
Note that all winding inductor pairs must be coupled with their own k cards in order for
the simulation to work properly. For a two-winding transformer, a single k card will sufﬁce.
For a three-winding transformer, three k cards must be speciﬁed (to link L1 with L2 , L2 with
L3 , and L1 with L3 ).
The L1 /L2 inductance ratio of 100:1 provides a 10:1 step-down voltage transformation ratio.
With 120 volts in we should see 12 volts out of the L2 winding. The L1 /L3 inductance ratio
of 100:25 (4:1) provides a 2:1 step-down voltage transformation ratio, which should give us 60
volts out of the L3 winding with 120 volts in.
Netlist:
transformer
v1 1 0 ac 120 sin
rbogus0 1 6 1e-3
l1 6 0 100
l2 2 4 1
l3 3 5 25
k1 l1 l2 0.999
k2 l2 l3 0.999
k3 l1 l3 0.999
r1 2 4 1000
r2 3 5 1000
rbogus1 5 0 1e10
rbogus2 4 0 1e10
.ac lin 1 60 60
.print ac v(1,0) v(2,0) v(3,0)
.end
105.
7.8. EXAMPLE CIRCUITS AND NETLISTS 97
Output:
freq v(1) v(2) v(3)
6.000E+01 1.200E+02 1.199E+01 5.993E+01
In this example, Rbogus0 is a very low-value resistor, serving to break up the source/inductor
loop of V1 /L1 . Rbogus1 and Rbogus2 are very high-value resistors necessary to provide DC paths
to ground on each of the isolated circuits. Note as well that one side of the primary circuit is
directly grounded. Without these ground references, SPICE will produce errors!
7.8.10 Full-wave bridge rectiﬁer
1 1
D1 D3
Rload
V1 15 V 2 + - 3
60 Hz 10 kΩ
D2 D4
0 0
Diodes, like all semiconductor components in SPICE, must be modeled so that SPICE knows
all the nitty-gritty details of how they’re supposed to work. Fortunately, SPICE comes with
a few generic models, and the diode is the most basic. Notice the .model card which simply
speciﬁes ”d” as the generic diode model for mod1. Again, since we’re plotting the waveforms
here, we need to specify all parameters of the AC source in a single card and print/plot all
values using the .tran option.
Netlist:
fullwave bridge rectifier
v1 1 0 sin(0 15 60 0 0)
rload 1 0 10k
d1 1 2 mod1
d2 0 2 mod1
d3 3 1 mod1
d4 3 0 mod1
.model mod1 d
.tran .5m 25m
.plot tran v(1,0) v(2,3)
.end
Output:
legend:
*: v(1)
+: v(2,3)
time v(1)
112.
104 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
To simulate an ideal operational ampliﬁer in SPICE, we use a voltage-dependent voltage source
as a differential ampliﬁer with extremely high gain. The ”e” card sets up the dependent voltage
source with four nodes, 3 and 0 for voltage output, and 1 and 0 for voltage input. No power
supply is needed for the dependent voltage source, unlike a real operational ampliﬁer. The
voltage gain is set at 999,000 in this case. The input voltage source (V1 ) sweeps from 0 to 3.5
volts in 0.05 volt steps.
Netlist:
Inverting opamp
v1 2 0 dc
e 3 0 0 1 999k
r1 3 1 3.29k
r2 1 2 1.18k
.dc v1 0 3.5 0.05
.print dc v(3,0)
.end
Output:
v1 v(3)
0.000E+00 0.000E+00
5.000E-02 -1.394E-01
1.000E-01 -2.788E-01
1.500E-01 -4.182E-01
2.000E-01 -5.576E-01
2.500E-01 -6.970E-01
3.000E-01 -8.364E-01
3.500E-01 -9.758E-01
4.000E-01 -1.115E+00
4.500E-01 -1.255E+00
5.000E-01 -1.394E+00
5.500E-01 -1.533E+00
6.000E-01 -1.673E+00
6.500E-01 -1.812E+00
7.000E-01 -1.952E+00
7.500E-01 -2.091E+00
8.000E-01 -2.231E+00
8.500E-01 -2.370E+00
9.000E-01 -2.509E+00
9.500E-01 -2.649E+00
1.000E+00 -2.788E+00
1.050E+00 -2.928E+00
1.100E+00 -3.067E+00
1.150E+00 -3.206E+00
1.200E+00 -3.346E+00
1.250E+00 -3.485E+00
1.300E+00 -3.625E+00
114.
106 CHAPTER 7. USING THE SPICE CIRCUIT SIMULATION PROGRAM
7.8.14 Noninverting op-amp circuit
R2 1 R1
3
10 kΩ 20 kΩ
0
1 −
2 (e) 3
2 +
Rbogus 10 kΩ
V1 5V
0 0
Another example of a SPICE quirk: since the dependent voltage source ”e” isn’t considered a
load to voltage source V1 , SPICE interprets V1 to be open-circuited and will refuse to analyze
it. The ﬁx is to connect Rbogus in parallel with V1 to act as a DC load. Being directly connected
across V1 , the resistance of Rbogus is not crucial to the operation of the circuit, so 10 kΩ will
work ﬁne. I decided not to sweep the V1 input voltage at all in this circuit for the sake of
keeping the netlist and output listing simple.
Netlist:
noninverting opamp
v1 2 0 dc 5
rbogus 2 0 10k
e 3 0 2 1 999k
r1 3 1 20k
r2 1 0 10k
.end
Output:
node voltage node voltage node voltage
( 1) 5.0000 ( 2) 5.0000 ( 3) 15.0000
115.
7.8. EXAMPLE CIRCUITS AND NETLISTS 107
7.8.15 Instrumentation ampliﬁer
1 1 + R3 R4
3 7 9
(e1)
2 − 10 kΩ 10 kΩ
Rbogus1 V1
0 to 10 V
R1 10 kΩ
0 0
2
7 −
Rgain 10 kΩ (e3) 9
8 +
5 Rload 10 kΩ
R2 10 kΩ 0
5 − R5 R6
4 (e2) 0
4 + 6 10 kΩ 8 10 kΩ
Rbogus2 V2 5V
0 0
Note the very high-resistance Rbogus1 and Rbogus2 resistors in the netlist (not shown in schematic
for brevity) across each input voltage source, to keep SPICE from thinking V1 and V2 were
open-circuited, just like the other op-amp circuit examples.
Netlist:
Instrumentation amplifier
v1 1 0
rbogus1 1 0 9e12
v2 4 0 dc 5
rbogus2 4 0 9e12
e1 3 0 1 2 999k
e2 6 0 4 5 999k
e3 9 0 8 7 999k
rload 9 0 10k
r1 2 3 10k
rgain 2 5 10k
r2 5 6 10k
r3 3 7 10k
r4 7 9 10k
r5 6 8 10k
r6 8 0 10k
.dc v1 0 10 1
.print dc v(9) v(3,6)
.end
Output:
122.
114 CHAPTER 8. TROUBLESHOOTING – THEORY AND PRACTICE
8.7 Potential pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.8 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.1
Perhaps the most valuable but difﬁcult-to-learn skill any technical person could have is the
ability to troubleshoot a system. For those unfamiliar with the term, troubleshooting means
the act of pinpointing and correcting problems in any kind of system. For an auto mechanic,
this means determining and ﬁxing problems in cars based on the car’s behavior. For a doctor,
this means correctly diagnosing a patient’s malady and prescribing a cure. For a business
expert, this means identifying the source(s) of inefﬁciency in a corporation and recommending
corrective measures.
Troubleshooters must be able to determine the cause or causes of a problem simply by
examining its effects. Rarely does the source of a problem directly present itself for all to see.
Cause/effect relationships are often complex, even for seemingly simple systems, and often
the proﬁcient troubleshooter is regarded by others as something of a miracle-worker for their
ability to quickly discern the root cause of a problem. While some people are gifted with a
natural talent for troubleshooting, it is a skill that can be learned like any other.
Sometimes the system to be analyzed is in so bad a state of affairs that there is no hope
of ever getting it working again. When investigators sift through the wreckage of a crashed
airplane, or when a doctor performs an autopsy, they must do their best to determine the
cause of massive failure after the fact. Fortunately, the task of the troubleshooter is usually
not this grim. Typically, a misbehaving system is still functioning to some degree and may
be stimulated and adjusted by the troubleshooter as part of the diagnostic procedure. In this
sense, troubleshooting is a lot like scientiﬁc method: determining cause/effect relationships by
means of live experimentation.
Like science, troubleshooting is a mixture of standard procedure and personal creativity.
There are certain procedures employed as tools to discern cause(s) from effects, but they are
impotent if not coupled with a creative and inquisitive mind. In the course of troubleshooting,
the troubleshooter may have to invent their own speciﬁc technique – adapted to the particular
system they’re working on – and/or modify tools to perform a special task. Creativity is nec-
essary in examining a problem from different perspectives: learning to ask different questions
when the ”standard” questions don’t lead to fruitful answers.
If there is one personality trait I’ve seen positively associated with excellent troubleshooting
more than any other, its technical curiosity. People fascinated by learning how things work,
and who aren’t discouraged by a challenging problem, tend to be better at troubleshooting than
others. Richard Feynman, the late physicist who taught at Caltech for many years, illustrates
to me the ultimate troubleshooting personality. Reading any of his (auto)biographical books
is both educating and entertaining, and I recommend them to anyone seeking to develop their
own scientiﬁc reasoning/troubleshooting skills.
123.
8.2. QUESTIONS TO ASK BEFORE PROCEEDING 115
8.2 Questions to ask before proceeding
• Has the system ever worked before? If yes, has anything happened to it since then that
could cause the problem?
• Has this system proven itself to be prone to certain types of failure?
• How urgent is the need for repair?
• What are the safety concerns, before I start troubleshooting?
• What are the process quality concerns, before I start troubleshooting (what can I do with-
out causing interruptions in production)?
These preliminary questions are not trivial. Indeed, they are essential to expedient and
safe troubleshooting. They are especially important when the system to be trouble-shot is
large, dangerous, and/or expensive.
Sometimes the troubleshooter will be required to work on a system that is still in full op-
eration (perhaps the ultimate example of this is a doctor diagnosing a live patient). Once the
cause or causes are determined to a high degree of certainty, there is the step of corrective ac-
tion. Correcting a system fault without signiﬁcantly interrupting the operation of the system
can be very challenging, and it deserves thorough planning.
When there is high risk involved in taking corrective action, such as is the case with per-
forming surgery on a patient or making repairs to an operating process in a chemical plant,
it is essential for the worker(s) to plan ahead for possible trouble. One question to ask before
proceeding with repairs is, ”how and at what point(s) can I abort the repairs if something goes
wrong?” In risky situations, it is vital to have planned ”escape routes” in your corrective action,
just in case things do not go as planned. A surgeon operating on a patient knows if there are
any ”points of no return” in such a procedure, and stops to re-check the patient before proceed-
ing past those points. He or she also knows how to ”back out” of a surgical procedure at those
points if needed.
8.3 General troubleshooting tips
When ﬁrst approaching a failed or otherwise misbehaving system, the new troubleshooter often
doesn’t know where to begin. The following strategies are not exhaustive by any means, but
provide the troubleshooter with a simple checklist of questions to ask in order to start isolating
the problem.
As tips, these troubleshooting suggestions are not comprehensive procedures: they serve
as starting points only for the troubleshooting process. An essential part of expedient trou-
bleshooting is probability assessment, and these tips help the troubleshooter determine which
possible points of failure are more or less likely than others. Final isolation of the system
failure is usually determined through more speciﬁc techniques (outlined in the next section –
Speciﬁc Troubleshooting Techniques).
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8.3.1 Prior occurrence
If this device or process has been historically known to fail in a certain particular way, and
the conditions leading to this common failure have not changed, check for this ”way” ﬁrst. A
corollary to this troubleshooting tip is the directive to keep detailed records of failure. Ideally,
a computer-based failure log is optimal, so that failures may be referenced by and correlated
to a number of factors such as time, date, and environmental conditions.
Example: The car’s engine is overheating. The last two times this happened, the cause was
low coolant level in the radiator.
What to do: Check the coolant level ﬁrst. Of course, past history by no means guarantees
the present symptoms are caused by the same problem, but since this is more likely, it makes
sense to check this ﬁrst.
If, however, the cause of routine failure in a system has been corrected (i.e. the leak causing
low coolant level in the past has been repaired), then this may not be a probable cause of
trouble this time.
8.3.2 Recent alterations
If a system has been having problems immediately after some kind of maintenance or other
change, the problems might be linked to those changes.
Example: The mechanic recently tuned my car’s engine, and now I hear a rattling noise
that I didn’t hear before I took the car in for repair.
What to do: Check for something that may have been left loose by the mechanic after his or
her tune-up work.
8.3.3 Function vs. non-function
If a system isn’t producing the desired end result, look for what it is doing correctly; in other
words, identify where the problem is not, and focus your efforts elsewhere. Whatever compo-
nents or subsystems necessary for the properly working parts to function are probably okay.
The degree of fault can often tell you what part of it is to blame.
Example: The radio works ﬁne on the AM band, but not on the FM band.
What to do: Eliminate from the list of possible causes, anything in the radio necessary for
the AM band’s function. Whatever the source of the problem is, it is speciﬁc to the FM band
and not to the AM band. This eliminates the audio ampliﬁer, speakers, fuse, power supply, and
almost all external wiring. Being able to eliminate sections of the system as possible failures
reduces the scope of the problem and makes the rest of the troubleshooting procedure more
efﬁcient.
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8.4. SPECIFIC TROUBLESHOOTING TECHNIQUES 117
8.3.4 Hypothesize
Based on your knowledge of how a system works, think of various kinds of failures that would
cause this problem (or these phenomena) to occur, and check for those failures (starting with
the most likely based on circumstances, history, or knowledge of component weaknesses).
Example: The car’s engine is overheating.
What to do: Consider possible causes for overheating, based on what you know of engine
operation. Either the engine is generating too much heat, or not getting rid of the heat well
enough (most likely the latter). Brainstorm some possible causes: a loose fan belt, clogged
radiator, bad water pump, low coolant level, etc. Investigate each one of those possibilities
before investigating alternatives.
8.4 Speciﬁc troubleshooting techniques
After applying some of the general troubleshooting tips to narrow the scope of a problem’s
location, there are techniques useful in further isolating it. Here are a few:
8.4.1 Swap identical components
In a system with identical or parallel subsystems, swap components between those subsystems
and see whether or not the problem moves with the swapped component. If it does, you’ve just
swapped the faulty component; if it doesn’t, keep searching!
This is a powerful troubleshooting method, because it gives you both a positive and a neg-
ative indication of the swapped component’s fault: when the bad part is exchanged between
identical systems, the formerly broken subsystem will start working again and the formerly
good subsystem will fail.
I was once able to troubleshoot an elusive problem with an automotive engine ignition
system using this method: I happened to have a friend with an automobile sharing the exact
same model of ignition system. We swapped parts between the engines (distributor, spark
plug wires, ignition coil – one at a time) until the problem moved to the other vehicle. The
problem happened to be a ”weak” ignition coil, and it only manifested itself under heavy load
(a condition that could not be simulated in my garage). Normally, this type of problem could
only be pinpointed using an ignition system analyzer (or oscilloscope) and a dynamometer
to simulate loaded driving conditions. This technique, however, conﬁrmed the source of the
problem with 100% accuracy, using no diagnostic equipment whatsoever.
Occasionally you may swap a component and ﬁnd that the problem still exists, but has
changed in some way. This tells you that the components you just swapped are somehow
different (different calibration, different function), and nothing more. However, don’t dismiss
this information just because it doesn’t lead you straight to the problem – look for other changes
in the system as a whole as a result of the swap, and try to ﬁgure out what these changes tell
you about the source of the problem.
An important caveat to this technique is the possibility of causing further damage. Suppose
a component has failed because of another, less conspicuous failure in the system. Swapping
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118 CHAPTER 8. TROUBLESHOOTING – THEORY AND PRACTICE
the failed component with a good component will cause the good component to fail as well. For
example, suppose that a circuit develops a short, which ”blows” the protective fuse for that
circuit. The blown fuse is not evident by inspection, and you don’t have a meter to electrically
test the fuse, so you decide to swap the suspect fuse with one of the same rating from a working
circuit. As a result of this, the good fuse that you move to the shorted circuit blows as well,
leaving you with two blown fuses and two non-working circuits. At least you know for certain
that the original fuse was blown, because the circuit it was moved to stopped working after the
swap, but this knowledge was gained only through the loss of a good fuse and the additional
”down time” of the second circuit.
Another example to illustrate this caveat is the ignition system problem previously men-
tioned. Suppose that the ”weak” ignition coil had caused the engine to backﬁre, damaging
the mufﬂer. If swapping ignition system components with another vehicle causes the prob-
lem to move to the other vehicle, damage may be done to the other vehicle’s mufﬂer as well.
As a general rule, the technique of swapping identical components should be used only when
there is minimal chance of causing additional damage. It is an excellent technique for isolating
non-destructive problems.
Example 1: You’re working on a CNC machine tool with X, Y, and Z-axis drives. The Y axis
is not working, but the X and Z axes are working. All three axes share identical components
(feedback encoders, servo motor drives, servo motors).
What to do: Exchange these identical components, one at a time, Y axis and either one of
the working axes (X or Z), and see after each swap whether or not the problem has moved with
the swap.
Example 2: A stereo system produces no sound on the left speaker, but the right speaker
works just ﬁne.
What to do: Try swapping respective components between the two channels and see if the
problem changes sides, from left to right. When it does, you’ve found the defective component.
For instance, you could swap the speakers between channels: if the problem moves to the other
side (i.e. the same speaker that was dead before is still dead, now that its connected to the right
channel cable) then you know that speaker is bad. If the problem stays on the same side (i.e.
the speaker formerly silent is now producing sound after having been moved to the other side
of the room and connected to the other cable), then you know the speakers are ﬁne, and the
problem must lie somewhere else (perhaps in the cable connecting the silent speaker to the
ampliﬁer, or in the ampliﬁer itself).
If the speakers have been veriﬁed as good, then you could check the cables using the same
method. Swap the cables so that each one now connects to the other channel of the ampliﬁer
and to the other speaker. Again, if the problem changes sides (i.e. now the right speaker is
now ”dead” and the left speaker now produces sound), then the cable now connected to the right
speaker must be defective. If neither swap (the speakers nor the cables) causes the problem
to change sides from left to right, then the problem must lie within the ampliﬁer (i.e. the left
channel output must be ”dead”).
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8.4. SPECIFIC TROUBLESHOOTING TECHNIQUES 119
8.4.2 Remove parallel components
If a system is composed of several parallel or redundant components which can be removed
without crippling the whole system, start removing these components (one at a time) and see
if things start to work again.
Example 1: A ”star” topology communications network between several computers has
failed. None of the computers are able to communicate with each other.
What to do: Try unplugging the computers, one at a time from the network, and see if
the network starts working again after one of them is unplugged. If it does, then that last
unplugged computer may be the one at fault (it may have been ”jamming” the network by
constantly outputting data or noise).
Example 2: A household fuse keeps blowing (or the breaker keeps tripping open) after a
short amount of time.
What to do: Unplug appliances from that circuit until the fuse or breaker quits interrupting
the circuit. If you can eliminate the problem by unplugging a single appliance, then that
appliance might be defective. If you ﬁnd that unplugging almost any appliance solves the
problem, then the circuit may simply be overloaded by too many appliances, neither of them
defective.
8.4.3 Divide system into sections and test those sections
In a system with multiple sections or stages, carefully measure the variables going in and out
of each stage until you ﬁnd a stage where things don’t look right.
Example 1: A radio is not working (producing no sound at the speaker))
What to do: Divide the circuitry into stages: tuning stage, mixing stages, ampliﬁer stage,
all the way through to the speaker(s). Measure signals at test points between these stages and
tell whether or not a stage is working properly.
Example 2: An analog summer circuit is not functioning properly.
Analog summer circuit
R 2R
−
R Vout
Vin1 +
R
Vin2
R
Vin3
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120 CHAPTER 8. TROUBLESHOOTING – THEORY AND PRACTICE
What to do: I would test the passive averager network (the three resistors at the lower-left
corner of the schematic) to see that the proper (averaged) voltage was seen at the noninverting
input of the op-amp. I would then measure the voltage at the inverting input to see if it was the
same as at the noninverting input (or, alternatively, measure the voltage difference between
the two inputs of the op-amp, as it should be zero). Continue testing sections of the circuit (or
just test points within the circuit) to see if you measure the expected voltages and currents.
8.4.4 Simplify and rebuild
Closely related to the strategy of dividing a system into sections, this is actually a design and
fabrication technique useful for new circuits, machines, or systems. It’s always easier begin
the design and construction process in little steps, leading to larger and larger steps, rather
than to build the whole thing at once and try to troubleshoot it as a whole.
Suppose that someone were building a custom automobile. He or she would be foolish to bolt
all the parts together without checking and testing components and subsystems as they went
along, expecting everything to work perfectly after its all assembled. Ideally, the builder would
check the proper operation of components along the way through the construction process:
start and tune the engine before its connected to the drivetrain, check for wiring problems
before all the cover panels are put in place, check the brake system in the driveway before
taking it out on the road, etc.
Countless times I’ve witnessed students build a complex experimental circuit and have
trouble getting it to work because they didn’t stop to check things along the way: test all
resistors before plugging them into place, make sure the power supply is regulating voltage
adequately before trying to power anything with it, etc. It is human nature to rush to comple-
tion of a project, thinking that such checks are a waste of valuable time. However, more time
will be wasted in troubleshooting a malfunctioning circuit than would be spent checking the
operation of subsystems throughout the process of construction.
Take the example of the analog summer circuit in the previous section for example: what
if it wasn’t working properly? How would you simplify it and test it in stages? Well, you
could reconnect the op-amp as a basic comparator and see if its responsive to differential input
voltages, and/or connect it as a voltage follower (buffer) and see if it outputs the same analog
voltage as what is input. If it doesn’t perform these simple functions, it will never perform its
function in the summer circuit! By stripping away the complexity of the summer circuit, paring
it down to an (almost) bare op-amp, you can test that component’s functionality and then build
from there (add resistor feedback and check for voltage ampliﬁcation, then add input resistors
and check for voltage summing), checking for expected results along the way.
8.4.5 Trap a signal
Set up instrumentation (such as a datalogger, chart recorder, or multimeter set on ”record”
mode) to monitor a signal over a period of time. This is especially helpful when tracking down
intermittent problems, which have a way of showing up the moment you’ve turned your back
and walked away.
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8.5. LIKELY FAILURES IN PROVEN SYSTEMS 121
This may be essential for proving what happens ﬁrst in a fast-acting system. Many fast
systems (especially shutdown ”trip” systems) have a ”ﬁrst out” monitoring capability to provide
this kind of data.
Example #1: A turbine control system shuts automatically in response to an abnormal con-
dition. By the time a technician arrives at the scene to survey the turbine’s condition, however,
everything is in a ”down” state and its impossible to tell what signal or condition was responsi-
ble for the initial shutdown, as all operating parameters are now ”abnormal.”
What to do: One technician I knew used a videocamera to record the turbine control panel,
so he could see what happened (by indications on the gauges) ﬁrst in an automatic-shutdown
event. Simply by looking at the panel after the fact, there was no way to tell which signal shut
the turbine down, but the videotape playback would show what happened in sequence, down
to a frame-by-frame time resolution.
Example #2: An alarm system is falsely triggering, and you suspect it may be due to a
speciﬁc wire connection going bad. Unfortunately, the problem never manifests itself while
you’re watching it!
What to do: Many modern digital multimeters are equipped with ”record” settings, whereby
they can monitor a voltage, current, or resistance over time and note whether that measure-
ment deviates substantially from a regular value. This is an invaluable tool for use in ”inter-
mittent” electronic system failures.
8.5 Likely failures in proven systems
The following problems are arranged in order from most likely to least likely, top to bottom.
This order has been determined largely from personal experience troubleshooting electrical
and electronic problems in automotive, industry, and home applications. This order also as-
sumes a circuit or system that has been proven to function as designed and has failed after
substantial operation time. Problems experienced in newly assembled circuits and systems do
not necessarily exhibit the same probabilities of occurrence.
8.5.1 Operator error
A frequent cause of system failure is error on the part of those human beings operating it.
This cause of trouble is placed at the top of the list, but of course the actual likelihood depends
largely on the particular individuals responsible for operation. When operator error is the
cause of a failure, it is unlikely that it will be admitted prior to investigation. I do not mean
to suggest that operators are incompetent and irresponsible – quite the contrary: these people
are often your best teachers for learning system function and obtaining a history of failure –
but the reality of human error cannot be overlooked. A positive attitude coupled with good
interpersonal skills on the part of the troubleshooter goes a long way in troubleshooting when
human error is the root cause of failure.
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8.5.2 Bad wire connections
As incredible as this may sound to the new student of electronics, a high percentage of electrical
and electronic system problems are caused by a very simple source of trouble: poor (i.e. open
or shorted) wire connections. This is especially true when the environment is hostile, including
such factors as high vibration and/or a corrosive atmosphere. Connection points found in any
variety of plug-and-socket connector, terminal strip, or splice are at the greatest risk for failure.
The category of ”connections” also includes mechanical switch contacts, which can be thought
of as a high-cycle connector. Improper wire termination lugs (such as a compression-style
connector crimped on the end of a solid wire – a deﬁnite faux pas) can cause high-resistance
connections after a period of trouble-free service.
It should be noted that connections in low-voltage systems tend to be far more troublesome
than connections in high-voltage systems. The main reason for this is the effect of arcing across
a discontinuity (circuit break) in higher-voltage systems tends to blast away insulating layers
of dirt and corrosion, and may even weld the two ends together if sustained long enough. Low-
voltage systems tend not to generate such vigorous arcing across the gap of a circuit break,
and also tend to be more sensitive to additional resistance in the circuit. Mechanical switch
contacts used in low-voltage systems beneﬁt from having the recommended minimum wetting
current conducted through them to promote a healthy amount of arcing upon opening, even if
this level of current is not necessary for the operation of other circuit components.
Although open failures tend to more common than shorted failures, ”shorts” still constitute
a substantial percentage of wiring failure modes. Many shorts are caused by degradation of
wire insulation. This, again, is especially true when the environment is hostile, including
such factors as high vibration, high heat, high humidity, or high voltage. It is rare to ﬁnd a
mechanical switch contact that is failed shorted, except in the case of high-current contacts
where contact ”welding” may occur in overcurrent conditions. Shorts may also be caused by
conductive buildup across terminal strip sections or the backs of printed circuit boards.
A common case of shorted wiring is the ground fault, where a conductor accidently makes
contact with either earth or chassis ground. This may change the voltage(s) present between
other conductors in the circuit and ground, thereby causing bizarre system malfunctions and/or
personnel hazard.
8.5.3 Power supply problems
These generally consist of tripped overcurrent protection devices or damage due to overheating.
Although power supply circuitry is usually less complex than the circuitry being powered, and
therefore should ﬁgure to be less prone to failure on that basis alone, it generally handles more
power than any other portion of the system and therefore must deal with greater voltages
and/or currents. Also, because of its relative design simplicity, a system’s power supply may
not receive the engineering attention it deserves, most of the engineering focus devoted to more
glamorous parts of the system.
8.5.4 Active components
Active components (ampliﬁcation devices) tend to fail with greater regularity than passive
(non-amplifying) devices, due to their greater complexity and tendency to amplify overvolt-
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8.6. LIKELY FAILURES IN UNPROVEN SYSTEMS 123
age/overcurrent conditions. Semiconductor devices are notoriously prone to failure due to elec-
trical transient (voltage/current surge) overloading and thermal (heat) overloading. Electron
tube devices are far more resistant to both of these failure modes, but are generally more prone
to mechanical failures due to their fragile construction.
8.5.5 Passive components
Non-amplifying components are the most rugged of all, their relative simplicity granting them
a statistical advantage over active devices. The following list gives an approximate relation of
failure probabilities (again, top being the most likely and bottom being the least likely):
• Capacitors (shorted), especially electrolytic capacitors. The paste electrolyte tends to lose
moisture with age, leading to failure. Thin dielectric layers may be punctured by over-
voltage transients.
• Diodes open (rectifying diodes) or shorted (Zener diodes).
• Inductor and transformer windings open or shorted to conductive core. Failures related
to overheating (insulation breakdown) are easily detected by smell.
• Resistors open, almost never shorted. Usually this is due to overcurrent heating, al-
though it is less frequently caused by overvoltage transient (arc-over) or physical damage
(vibration or impact). Resistors may also change resistance value if overheated!
8.6 Likely failures in unproven systems
”All men are liable to error;”
John Locke
Whereas the last section deals with component failures in systems that have been success-
fully operating for some time, this section concentrates on the problems plaguing brand-new
systems. In this case, failure modes are generally not of the aging kind, but are related to
mistakes in design and assembly caused by human beings.
8.6.1 Wiring problems
In this case, bad connections are usually due to assembly error, such as connection to the wrong
point or poor connector fabrication. Shorted failures are also seen, but usually involve miscon-
nections (conductors inadvertently attached to grounding points) or wires pinched under box
covers.
Another wiring-related problem seen in new systems is that of electrostatic or electromag-
netic interference between different circuits by way of close wiring proximity. This kind of
problem is easily created by routing sets of wires too close to each other (especially routing
signal cables close to power conductors), and tends to be very difﬁcult to identify and locate
with test equipment.
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124 CHAPTER 8. TROUBLESHOOTING – THEORY AND PRACTICE
8.6.2 Power supply problems
Blown fuses and tripped circuit breakers are likely sources of trouble, especially if the project in
question is an addition to an already-functioning system. Loads may be larger than expected,
resulting in overloading and subsequent failure of power supplies.
8.6.3 Defective components
In the case of a newly-assembled system, component fault probabilities are not as predictable
as in the case of an operating system that fails with age. Any type of component – active or
passive – may be found defective or of imprecise value ”out of the box” with roughly equal prob-
ability, barring any speciﬁc sensitivities in shipping (i.e fragile vacuum tubes or electrostati-
cally sensitive semiconductor components). Moreover, these types of failures are not always as
easy to identify by sight or smell as an age- or transient-induced failure.
8.6.4 Improper system conﬁguration
Increasingly seen in large systems using microprocessor-based components, ”programming”
issues can still plague non-microprocessor systems in the form of incorrect time-delay relay
settings, limit switch calibrations, and drum switch sequences. Complex components having
conﬁguration ”jumpers” or switches to control behavior may not be ”programmed” properly.
Components may be used in a new system outside of their tolerable ranges. Resistors, for
example, with too low of power ratings, of too great of tolerance, may have been installed.
Sensors, instruments, and controlling mechanisms may be uncalibrated, or calibrated to the
wrong ranges.
8.6.5 Design error
Perhaps the most difﬁcult to pinpoint and the slowest to be recognized (especially by the chief
designer) is the problem of design error, where the system fails to function simply because
it cannot function as designed. This may be as trivial as the designer specifying the wrong
components in a system, or as fundamental as a system not working due to the designer’s
improper knowledge of physics.
I once saw a turbine control system installed that used a low-pressure switch on the lubri-
cation oil tubing to shut down the turbine if oil pressure dropped to an insufﬁcient level. The
oil pressure for lubrication was supplied by an oil pump turned by the turbine. When installed,
the turbine refused to start. Why? Because when it was stopped, the oil pump was not turning,
thus there was no oil pressure to lubricate the turbine. The low-oil-pressure switch detected
this condition and the control system maintained the turbine in shutdown mode, preventing
it from starting. This is a classic example of a design ﬂaw, and it could only be corrected by a
change in the system logic.
While most design ﬂaws manifest themselves early in the operational life of the system,
some remain hidden until just the right conditions exist to trigger the fault. These types of
ﬂaws are the most difﬁcult to uncover, as the troubleshooter usually overlooks the possibility
of design error due to the fact that the system is assumed to be ”proven.” The example of the
turbine lubrication system was a design ﬂaw impossible to ignore on start-up. An example of
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8.7. POTENTIAL PITFALLS 125
a ”hidden” design ﬂaw might be a faulty emergency coolant system for a machine, designed to
remain inactive until certain abnormal conditions are reached – conditions which might never
be experienced in the life of the system.
8.7 Potential pitfalls
Fallacious reasoning and poor interpersonal relations account for more failed or belabored trou-
bleshooting efforts than any other impediments. With this in mind, the aspiring troubleshooter
needs to be familiar with a few common troubleshooting mistakes.
Trusting that a brand-new component will always be good. While it is generally true
that a new component will be in good condition, it is not always true. It is also possible that
a component has been mis-labeled and may have the wrong value (usually this mis-labeling is
a mistake made at the point of distribution or warehousing and not at the manufacturer, but
again, not always!).
Not periodically checking your test equipment. This is especially true with battery-
powered meters, as weak batteries may give spurious readings. When using meters to safety-
check for dangerous voltage, remember to test the meter on a known source of voltage both
before and after checking the circuit to be serviced, to make sure the meter is in proper operat-
ing condition.
Assuming there is only one failure to account for the problem. Single-failure sys-
tem problems are ideal for troubleshooting, but sometimes failures come in multiple numbers.
In some instances, the failure of one component may lead to a system condition that damages
other components. Sometimes a component in marginal condition goes undetected for a long
time, then when another component fails the system suffers from problems with both compo-
nents.
Mistaking coincidence for causality. Just because two events occurred at nearly the
same time does not necessarily mean one event caused the other! They may be both conse-
quences of a common cause, or they may be totally unrelated! If possible, try to duplicate the
same condition suspected to be the cause and see if the event suspected to be the coincidence
happens again. If not, then there is either no causal relationship as assumed. This may mean
there is no causal relationship between the two events whatsoever, or that there is a causal
relationship, but just not the one you expected.
Self-induced blindness. After a long effort at troubleshooting a difﬁcult problem, you
may become tired and begin to overlook crucial clues to the problem. Take a break and let
someone else look at it for a while. You will be amazed at what a difference this can make.
On the other hand, it is generally a bad idea to solicit help at the start of the troubleshooting
process. Effective troubleshooting involves complex, multi-level thinking, which is not easily
communicated with others. More often than not, ”team troubleshooting” takes more time and
causes more frustration than doing it yourself. An exception to this rule is when the knowledge
of the troubleshooters is complementary: for example, a technician who knows electronics
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126 CHAPTER 8. TROUBLESHOOTING – THEORY AND PRACTICE
but not machine operation, teamed with an operator who knows machine function but not
electronics.
Failing to question the troubleshooting work of others on the same job. This may
sound rather cynical and misanthropic, but it is sound scientiﬁc practice. Because it is easy to
overlook important details, troubleshooting data received from another troubleshooter should
be personally veriﬁed before proceeding. This is a common situation when troubleshooters
”change shifts” and a technician takes over for another technician who is leaving before the
job is done. It is important to exchange information, but do not assume the prior techni-
cian checked everything they said they did, or checked it perfectly. I’ve been hindered in my
troubleshooting efforts on many occasions by failing to verify what someone else told me they
checked.
Being pressured to ”hurry up.” When an important system fails, there will be pressure
from other people to ﬁx the problem as quickly as possible. As they say in business, ”time
is money.” Having been on the receiving end of this pressure many times, I can understand
the need for expedience. However, in many cases there is a higher priority: caution. If the
system in question harbors great danger to life and limb, the pressure to ”hurry up” may
result in injury or death. At the very least, hasty repairs may result in further damage when
the system is restarted. Most failures can be recovered or at least temporarily repaired in
short time if approached intelligently. Improper ”ﬁxes” resulting in haste often lead to damage
that cannot be recovered in short time, if ever. If the potential for greater harm is present, the
troubleshooter needs to politely address the pressure received from others, and maintain their
perspective in the midst of chaos. Interpersonal skills are just as important in this realm as
technical ability!
Finger-pointing. It is all too easy to blame a problem on someone else, for reasons of
ignorance, pride, laziness, or some other unfortunate facet of human nature. When the respon-
sibility for system maintenance is divided into departments or work crews, troubleshooting
efforts are often hindered by blame cast between groups. ”It’s a mechanical problem . . . its
an electrical problem . . . its an instrument problem . . .” ad inﬁnitum, ad nauseum, is all too
common in the workplace. I have found that a positive attitude does more to quench the ﬁres
of blame than anything else.
On one particular job, I was summoned to ﬁx a problem in a hydraulic system assumed to
be related to the electronic metering and controls. My troubleshooting isolated the source of
trouble to a faulty control valve, which was the domain of the millwright (mechanical) crew. I
knew that the millwright on shift was a contentious person, so I expected trouble if I simply
passed the problem on to his department. Instead, I politely explained to him and his super-
visor the nature of the problem as well as a brief synopsis of my reasoning, then proceeded
to help him replace the faulty valve, even though it wasn’t ”my” responsibility to do so. As a
result, the problem was ﬁxed very quickly, and I gained the respect of the millwright.
8.8 Contributors
Contributors to this chapter are listed in chronological order of their contributions, from most
recent to ﬁrst. See Appendix 2 (Contributor List) for dates and contact information.
135.
8.8. CONTRIBUTORS 127
Alejandro Gamero Divasto (January 2002): contributed troubleshooting tips regarding
potential hazards of swapping two similar components, avoiding pressure placed on the trou-
bleshooter, perils of ”team” troubleshooting, wisdom of recording system history, operator error
as a cause of failure, and the perils of ﬁnger-pointing.
136.
128 CHAPTER 8. TROUBLESHOOTING – THEORY AND PRACTICE
138.
130 CHAPTER 9. CIRCUIT SCHEMATIC SYMBOLS
9.1 Wires and connections
Older convention
Connected Not connected
Newer convention
Connected Not connected
Older electrical schematics showed connecting wires crossing, while non-connecting wires ”jumped”
over each other with little half-circle marks. Newer electrical schematics show connecting
wires joining with a dot, while non-connecting wires cross with no dot. However, some peo-
ple still use the older convention of connecting wires crossing with no dot, which may create
confusion.
For this reason, I opt to use a hybrid convention, with connecting wires unambiguously
connected by a dot, and non-connecting wires unambiguously ”jumping” over one another with
a half-circle mark. While this may be frowned upon by some, it leaves no room for interpreta-
tional error: in each case, the intent is clear and unmistakable:
Convention used in this book
Connected Not connected
139.
9.2. POWER SOURCES 131
9.2 Power sources
DC voltage DC voltage AC voltage
+
−
Variable
DC voltage DC current
A diagonal arrow +
represents variability
for any component! -
Generator AC current
Gen
9.3 Resistors
Fixed-value Rheostat
Potentiometer Tapped Thermistor
to
Photoresistor
142.
134 CHAPTER 9. CIRCUIT SCHEMATIC SYMBOLS
9.7 Switches, hand actuated
SPST toggle Pushbutton
normally open DPST toggle normally open
SPST toggle Pushbutton
normally closed normally closed
DPDT toggle
SPDT toggle SPST joystick
position of dot
on circle indicates
joystick direction
4PDT toggle
143.
9.8. SWITCHES, PROCESS ACTUATED 135
9.8 Switches, process actuated
Normally open shown on top; normally closed on bottom
Level Pressure Flow Temperature
F
Electronic R
Limit Limit Speed
F
R
It is very important to keep in mind that the ”normal” contact status of a process-actuated
switch refers to its status when the process is absent and/or inactive, not ”normal” in the sense
of process conditions as expected during routine operation. For instance, a normally-closed
low-ﬂow detection switch installed on a coolant pipe will be maintained in the actuated state
(open) when there is regular coolant ﬂow through the pipe. If the coolant ﬂow stops, the ﬂow
switch will go to its ”normal” (unactuated) status of closed.
A limit switch is one actuated by contact with a moving machine part. An electronic limit
switch senses mechanical motion, but does so using light, magnetic ﬁelds, or other non-contact
means.
145.
9.11. DIODES 137
9.11 Diodes
Generic Schottky Shockley Constant current
A K A K A K A K
Zener Light-emitting Photo- Step recovery
A K A K A K A K
Tunnel Varactor PIN Vacuum tube
P
A K A K A K
A = Anode
K = Cathode
C
H1 H2
146.
138 CHAPTER 9. CIRCUIT SCHEMATIC SYMBOLS
9.12 Transistors, bipolar
. . . with case
Bipolar NPN Bipolar PNP
B B
E C E C
Photo-
Dual-emitter NPN Dual-emitter PNP
B B
E C E1 C E1 C
E2 E2
Darlington pair E = Emitter Sziklai pair
B B
B = Base
C = Collector
C C
E E
9.13 Transistors, junction ﬁeld-effect (JFET)
N-channel P-channel . . . with case
G G
S D S D
S = Source
G = Gate
D = Drain
147.
9.14. TRANSISTORS, INSULATED-GATE FIELD-EFFECT (IGFET OR MOSFET) 139
9.14 Transistors, insulated-gate ﬁeld-effect (IGFET or MOS-
FET)
N-channel P-channel N-channel P-channel
depletion depletion enhancement enhancement
G G G G
S D S D S D S D
SS SS SS SS
N-channel P-channel N-channel P-channel
depletion depletion enhancement enhancement
G G G G
S D S D S D S D
S = Source . . . with case
G = Gate
D = Drain
SS = Substrate
9.15 Transistors, hybrid
. . . with case
IGBT (NPN) IGBT (PNP)
G G
E C E C
. . . with case
IGBT (N-channel) IGBT (P-channel)
G G
E C E C
E = Emitter
G = Gate
C = Collector
148.
140 CHAPTER 9. CIRCUIT SCHEMATIC SYMBOLS
9.16 Thyristors
Shockley DIAC SCR LASCR
A K A K A K
G G
TRIAC Opto-TRIAC GA SCS GCS
MT2 MT1 MT2 MT1 A K A K
G G GK G
GTO UJT B1 A = Anode
A K
E K = Cathode
G G = Gate
B2
MT = Main Terminal
E = Emitter
B = Base
149.
9.17. INTEGRATED CIRCUITS 141
9.17 Integrated circuits
Operational amplifier (alternative) Norton op-amp
− - −
+ + +
Inverter AND gate OR gate XOR gate
Inverter NAND gate NOR gate XNOR gate
Negative-AND Negative-OR
Buffer gate gate
Gate with open- Gate with Schmitt
collector output trigger input
150.
142 CHAPTER 9. CIRCUIT SCHEMATIC SYMBOLS
S-R Latch Enabled S-R Latch S-R Flip-flop
S Q S Q S Q
E C
R Q R Q R Q
D Latch D Flip-flop J-K Flip-flop
D Q D Q J Q
E C C
Q Q K Q
152.
144 CHAPTER 9. CIRCUIT SCHEMATIC SYMBOLS
9.18 Electron tubes
Diode Glow tube Phototube
P P C
C C A
H1 H2
Triode Tetrode Beam tetrode
P P P
S
G S
G G
C C C
H1 H2 H1 H2 H1 H2
Pentode Pentode Thyratron
P P P
Sup
S S G
G G
C C C
H1 H2 H1 H2 H1 H2
Ignitron Cathode Ray Tube
A
I H V
C
P = Plate S = Screen
G = Grid A = Anode
C = Cathode H = Heater
I = Ignitor Sup = Suppressor
153.
Chapter 10
PERIODIC TABLE OF THE
ELEMENTS
Contents
10.1 Table (landscape view) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
10.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
10.1 Table (landscape view)
See Figure 10.1.
10.2 Data
Atomic masses shown in parentheses indicate the most stable isotope (longest half-life) known.
Electron conﬁguration data was taken from Douglas C. Giancoli’s Physics, 3rd edition. Av-
erage atomic masses were taken from Kenneth W. Whitten’s, Kenneth D. Gailey’s, and Ray-
mond E. Davis’ General Chemistry, 3rd edition. In the latter book, the masses were speciﬁed
as 1985 IUPAC values.
145
154.
146
1 IA 13 VIIIA
H 1 He 2
Hydrogen
1.00794
Periodic Table of the Elements Helium
4.00260
Group old Metalloids Nonmetals
1s1 2 IIA Group new 1 IA 13 IIIA 14 IVA 15 VA 16 VIA 17 VIIA 1s2
Li 3 Be 4 Symbol K 19 Atomic number B 5 C 6 N 7 O 8 F 9 Ne 10
Lithium Beryllium Potassium Boron Carbon Nitrogen Oxygen Fluorine Neon
Figure 10.1: Periodic table of chemical elements.
6.941 9.012182 Name 39.0983 10.81 12.011 14.0067 15.9994 18.9984 20.179
Atomic mass
2s1 2s2 4s1 (averaged according to 2p1 2p2 2p3 2p4 2p5 2p6
Electron occurence on earth)
Na 11 Mg 12 configuration Al 13 Si 14 P 15 S 16 Cl 17 Ar 18
Sodium Magnesium Aluminum Silicon Phosphorus Sulfur Chlorine Argon
22.989768 24.3050 26.9815 28.0855 30.9738 32.06 35.453 39.948
Metals
3s1 3s2 3 IIIB 4 IVB 5 VB 6 VIB 7 VIIB 8 VIIIB 9 VIIIB 10 VIIIB 11 IB 12 IIB 3p1 3p2 3p3 3p4 3p5 3p6
K 19 Ca 20 Sc 21 Ti 22 V 23 Cr 24 Mn 25 Fe 26 Co 27 Ni 28 Cu 29 Zn 30 Ga 31 Ge 32 As 33 Se 34 Br 35 Kr 36
Potassium Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton
CHAPTER 10. PERIODIC TABLE OF THE ELEMENTS
39.0983 40.078 44.955910 47.88 50.9415 51.9961 54.93805 55.847 58.93320 58.69 63.546 65.39 69.723 72.61 74.92159 78.96 79.904 83.80
4s1 4s2 3d14s2 3d24s2 3d34s2 3d54s1 3d54s2 3d64s2 3d74s2 3d84s2 3d104s1 3d104s2 4p1 4p2 4p3 4p4 4p5 4p6
Rb 37 Sr 38 Y 39 Zr 40 Nb 41 Mo 42 Tc 43 Ru 44 Rh 45 Pd 46 Ag 47 Cd 48 In 49 Sn 50 Sb 51 Te 52 I 53 Xe 54
Rubidium Strontium Yttrium Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium Palladium Silver Cadmium Indium Tin Antimony Tellurium Iodine Xenon
85.4678 87.62 88.90585 91.224 92.90638 95.94 (98) 101.07 102.90550 106.42 107.8682 112.411 114.82 118.710 121.75 127.60 126.905 131.30
5s1 5s2 4d15s2 4d25s2 4d45s1 4d55s1 4d55s2 4d75s1 4d85s1 4d105s0 4d105s1 4d105s2 5p1 5p2 5p3 5p4 5p5 5p6
Cs 55 Ba 56 57 - 71 Hf 72 Ta 73 W 74 Re 75 Os 76 Ir 77 Pt 78 Au 79 Hg 80 Tl 81 Pb 82 Bi 83 Po 84 At 85 Rn 86
Cesium Barium Lanthanide Hafnium Tantalum Tungsten Rhenium Osmium Iridium Platinum Gold Mercury Thallium Lead Bismuth Polonium Astatine Radon
132.90543 137.327 series 178.49 180.9479 183.85 186.207 190.2 192.22 195.08 196.96654 200.59 204.3833 207.2 208.98037 (209) (210) (222)
6s1 6s2 5d26s2 5d36s2 5d46s2 5d56s2 5d66s2 5d76s2 5d96s1 5d106s1 5d106s2 6p1 6p2 6p3 6p4 6p5 6p6
Fr 87 Ra 88 89 - 103 Unq 104 Unp 105 Unh 106 Uns 107 108 109
Francium Radium Actinide Unnilquadium Unnilpentium Unnilhexium Unnilseptium
(223) (226) series (261) (262) (263) (262)
1 2 2 2 3 2 4 2
7s 7s 6d 7s 6d 7s 6d 7s
La 57 Ce 58 Pr 59 Nd 60 Pm 61 Sm 62 Eu 63 Gd 64 Tb 65 Dy 66 Ho 67 Er 68 Tm 69 Yb 70 Lu 71
Lanthanide Lanthanum Cerium Praseodymium Neodymium Promethium Samarium Europium Gadolinium Terbium Dysprosium Holmium Erbium Thulium Ytterbium Lutetium
series 138.9055 140.115 140.90765 144.24 (145) 150.36 151.965 157.25 158.92534 162.50 164.93032 167.26 168.93421 173.04 174.967
5d16s2 4f15d16s2 4f36s2 4f46s2 4f56s2 4f66s2 4f76s2 4f75d16s2 4f96s2 4f106s2 4f116s2 4f126s2 4f136s2 4f146s2 4f145d16s2
Ac 89 Th 90 Pa 91 U 92 Np 93 Pu 94 Am 95 Cm 96 Bk 97 Cf 98 Es 99 Fm 100 Md 101 No 102 Lr 103
Actinide Actinium Thorium Protactinium Uranium Neptunium Plutonium Americium Curium Berkelium Californium Einsteinium Fermium Mendelevium Nobelium Lawrencium
series (227) 232.0381 231.03588 238.0289 (237) (244) (243) (247) (247) (251) (252) (257) (258) (259) (260)
6d17s2 6d27s2 5f26d17s2 5f36d17s2 5f46d17s2 5f66d07s2 5f76d07s2 5f76d17s2 5f96d07s2 5f106d07s2 5f116d07s2 5f126d07s2 5f136d07s2 6d07s2 6d17s2
155.
Appendix A-1
ABOUT THIS BOOK
A-1.1 Purpose
They say that necessity is the mother of invention. At least in the case of this book, that adage
is true. As an industrial electronics instructor, I was forced to use a sub-standard textbook
during my ﬁrst year of teaching. My students were daily frustrated with the many typograph-
ical errors and obscure explanations in this book, having spent much time at home struggling
to comprehend the material within. Worse yet were the many incorrect answers in the back of
the book to selected problems. Adding insult to injury was the $100+ price.
Contacting the publisher proved to be an exercise in futility. Even though the particular
text I was using had been in print and in popular use for a couple of years, they claimed my
complaint was the ﬁrst they’d ever heard. My request to review the draft for the next edition
of their book was met with disinterest on their part, and I resolved to ﬁnd an alternative text.
Finding a suitable alternative was more difﬁcult than I had imagined. Sure, there were
plenty of texts in print, but the really good books seemed a bit too heavy on the math and the
less intimidating books omitted a lot of information I felt was important. Some of the best
books were out of print, and those that were still being printed were quite expensive.
It was out of frustration that I compiled Lessons in Electric Circuits from notes and ideas I
had been collecting for years. My primary goal was to put readable, high-quality information
into the hands of my students, but a secondary goal was to make the book as affordable as
possible. Over the years, I had experienced the beneﬁt of receiving free instruction and encour-
agement in my pursuit of learning electronics from many people, including several teachers
of mine in elementary and high school. Their selﬂess assistance played a key role in my own
studies, paving the way for a rewarding career and fascinating hobby. If only I could extend
the gift of their help by giving to other people what they gave to me . . .
So, I decided to make the book freely available. More than that, I decided to make it ”open,”
following the same development model used in the making of free software (most notably the
various UNIX utilities released by the Free Software Foundation, and the Linux operating
147
156.
148 APPENDIX A-1. ABOUT THIS BOOK
system, whose fame is growing even as I write). The goal was to copyright the text – so as to
protect my authorship – but expressly allow anyone to distribute and/or modify the text to suit
their own needs with a minimum of legal encumbrance. This willful and formal revoking of
standard distribution limitations under copyright is whimsically termed copyleft. Anyone can
”copyleft” their creative work simply by appending a notice to that effect on their work, but
several Licenses already exist, covering the ﬁne legal points in great detail.
The ﬁrst such License I applied to my work was the GPL – General Public License – of the
Free Software Foundation (GNU). The GPL, however, is intended to copyleft works of computer
software, and although its introductory language is broad enough to cover works of text, its
wording is not as clear as it could be for that application. When other, less speciﬁc copyleft
Licenses began appearing within the free software community, I chose one of them (the Design
Science License, or DSL) as the ofﬁcial notice for my project.
In ”copylefting” this text, I guaranteed that no instructor would be limited by a text insufﬁ-
cient for their needs, as I had been with error-ridden textbooks from major publishers. I’m sure
this book in its initial form will not satisfy everyone, but anyone has the freedom to change it,
leveraging my efforts to suit variant and individual requirements. For the beginning student
of electronics, learn what you can from this book, editing it as you feel necessary if you come
across a useful piece of information. Then, if you pass it on to someone else, you will be giving
them something better than what you received. For the instructor or electronics professional,
feel free to use this as a reference manual, adding or editing to your heart’s content. The
only ”catch” is this: if you plan to distribute your modiﬁed version of this text, you must give
credit where credit is due (to me, the original author, and anyone else whose modiﬁcations are
contained in your version), and you must ensure that whoever you give the text to is aware of
their freedom to similarly share and edit the text. The next chapter covers this process in more
detail.
It must be mentioned that although I strive to maintain technical accuracy in all of this
book’s content, the subject matter is broad and harbors many potential dangers. Electricity
maims and kills without provocation, and deserves the utmost respect. I strongly encourage
experimentation on the part of the reader, but only with circuits powered by small batteries
where there is no risk of electric shock, ﬁre, explosion, etc. High-power electric circuits should
be left to the care of trained professionals! The Design Science License clearly states that
neither I nor any contributors to this book bear any liability for what is done with its contents.
A-1.2 The use of SPICE
One of the best ways to learn how things work is to follow the inductive approach: to observe
speciﬁc instances of things working and derive general conclusions from those observations.
In science education, labwork is the traditionally accepted venue for this type of learning, al-
though in many cases labs are designed by educators to reinforce principles previously learned
through lecture or textbook reading, rather than to allow the student to learn on their own
through a truly exploratory process.
Having taught myself most of the electronics that I know, I appreciate the sense of frustra-
tion students may have in teaching themselves from books. Although electronic components
are typically inexpensive, not everyone has the means or opportunity to set up a laboratory
in their own homes, and when things go wrong there’s no one to ask for help. Most textbooks
157.
A-1.3. ACKNOWLEDGEMENTS 149
seem to approach the task of education from a deductive perspective: tell the student how
things are supposed to work, then apply those principles to speciﬁc instances that the student
may or may not be able to explore by themselves. The inductive approach, as useful as it is, is
hard to ﬁnd in the pages of a book.
However, textbooks don’t have to be this way. I discovered this when I started to learn a
computer program called SPICE. It is a text-based piece of software intended to model circuits
and provide analyses of voltage, current, frequency, etc. Although nothing is quite as good as
building real circuits to gain knowledge in electronics, computer simulation is an excellent al-
ternative. In learning how to use this powerful tool, I made a discovery: SPICE could be used
within a textbook to present circuit simulations to allow students to ”observe” the phenomena
for themselves. This way, the readers could learn the concepts inductively (by interpreting
SPICE’s output) as well as deductively (by interpreting my explanations). Furthermore, in
seeing SPICE used over and over again, they should be able to understand how to use it them-
selves, providing a perfectly safe means of experimentation on their own computers with circuit
simulations of their own design.
Another advantage to including computer analyses in a textbook is the empirical veriﬁ-
cation it adds to the concepts presented. Without demonstrations, the reader is left to take
the author’s statements on faith, trusting that what has been written is indeed accurate. The
problem with faith, of course, is that it is only as good as the authority in which it is placed and
the accuracy of interpretation through which it is understood. Authors, like all human beings,
are liable to err and/or communicate poorly. With demonstrations, however, the reader can
immediately see for themselves that what the author describes is indeed true. Demonstrations
also serve to clarify the meaning of the text with concrete examples.
SPICE is introduced early in volume I (DC) of this book series, and hopefully in a gentle
enough way that it doesn’t create confusion. For those wishing to learn more, a chapter in this
volume (volume V) contains an overview of SPICE with many example circuits. There may
be more ﬂashy (graphic) circuit simulation programs in existence, but SPICE is free, a virtue
complementing the charitable philosophy of this book very nicely.
A-1.3 Acknowledgements
First, I wish to thank my wife, whose patience during those many and long evenings (and
weekends!) of typing has been extraordinary.
I also wish to thank those whose open-source software development efforts have made this
endeavor all the more affordable and pleasurable. The following is a list of various free com-
puter software used to make this book, and the respective programmers:
• GNU/Linux Operating System – Linus Torvalds, Richard Stallman, and a host of others
too numerous to mention.
• Vim text editor – Bram Moolenaar and others.
• Xcircuit drafting program – Tim Edwards.
• SPICE circuit simulation program – too many contributors to mention.
• TEX text processing system – Donald Knuth and others.
158.
150 APPENDIX A-1. ABOUT THIS BOOK
• Texinfo document formatting system – Free Software Foundation.
• LTEX document formatting system – Leslie Lamport and others.
A
• Gimp image manipulation program – too many contributors to mention.
Appreciation is also extended to Robert L. Boylestad, whose ﬁrst edition of Introductory
Circuit Analysis taught me more about electric circuits than any other book. Other important
texts in my electronics studies include the 1939 edition of The ”Radio” Handbook, Bernard
Grob’s second edition of Introduction to Electronics I, and Forrest Mims’ original Engineer’s
Notebook.
Thanks to the staff of the Bellingham Antique Radio Museum, who were generous enough
to let me terrorize their establishment with my camera and ﬂash unit.
I wish to speciﬁcally thank Jeffrey Elkner and all those at Yorktown High School for being
willing to host my book as part of their Open Book Project, and to make the ﬁrst effort in con-
tributing to its form and content. Thanks also to David Sweet (website: (http://www.andamooka.org))
and Ben Crowell (website: (http://www.lightandmatter.com)) for providing encourage-
ment, constructive criticism, and a wider audience for the online version of this book.
Thanks to Michael Stutz for drafting his Design Science License, and to Richard Stallman
for pioneering the concept of copyleft.
Last but certainly not least, many thanks to my parents and those teachers of mine who
saw in me a desire to learn about electricity, and who kindled that ﬂame into a passion for
discovery and intellectual adventure. I honor you by helping others as you have helped me.
Tony Kuphaldt, July 2001
”A candle loses nothing of its light when lighting another”
Kahlil Gibran
159.
Appendix A-2
CONTRIBUTOR LIST
A-2.1 How to contribute to this book
As a copylefted work, this book is open to revision and expansion by any interested parties.
The only ”catch” is that credit must be given where credit is due. This is a copyrighted work:
it is not in the public domain!
If you wish to cite portions of this book in a work of your own, you must follow the same
guidelines as for any other copyrighted work. Here is a sample from the Design Science Li-
cense:
The Work is copyright the Author. All rights to the Work are reserved
by the Author, except as specifically described below. This License
describes the terms and conditions under which the Author permits you
to copy, distribute and modify copies of the Work.
In addition, you may refer to the Work, talk about it, and (as
dictated by "fair use") quote from it, just as you would any
copyrighted material under copyright law.
Your right to operate, perform, read or otherwise interpret and/or
execute the Work is unrestricted; however, you do so at your own risk,
because the Work comes WITHOUT ANY WARRANTY -- see Section 7 ("NO
WARRANTY") below.
If you wish to modify this book in any way, you must document the nature of those modiﬁ-
cations in the ”Credits” section along with your name, and ideally, information concerning how
you may be contacted. Again, the Design Science License:
Permission is granted to modify or sample from a copy of the Work,
151
160.
152 APPENDIX A-2. CONTRIBUTOR LIST
producing a derivative work, and to distribute the derivative work
under the terms described in the section for distribution above,
provided that the following terms are met:
(a) The new, derivative work is published under the terms of this
License.
(b) The derivative work is given a new name, so that its name or
title can not be confused with the Work, or with a version of
the Work, in any way.
(c) Appropriate authorship credit is given: for the differences
between the Work and the new derivative work, authorship is
attributed to you, while the material sampled or used from
the Work remains attributed to the original Author; appropriate
notice must be included with the new work indicating the nature
and the dates of any modifications of the Work made by you.
Given the complexities and security issues surrounding the maintenance of ﬁles comprising
this book, it is recommended that you submit any revisions or expansions to the original author
(Tony R. Kuphaldt). You are, of course, welcome to modify this book directly by editing your
own personal copy, but we would all stand to beneﬁt from your contributions if your ideas were
incorporated into the online “master copy” where all the world can see it.
A-2.2 Credits
All entries arranged in alphabetical order of surname. Major contributions are listed by indi-
vidual name with some detail on the nature of the contribution(s), date, contact info, etc. Minor
contributions (typo corrections, etc.) are listed by name only for reasons of brevity. Please un-
derstand that when I classify a contribution as “minor,” it is in no way inferior to the effort
or value of a “major” contribution, just smaller in the sense of less text changed. Any and all
contributions are gratefully accepted. I am indebted to all those who have given freely of their
own knowledge, time, and resources to make this a better book!
A-2.2.1 Dennis Crunkilton
• Date(s) of contribution(s):October 2005 to present
• Nature of contribution:Ch 1, added permitivity, capacitor and inductor formulas, wire
table; 10/2005.
• Nature of contribution:Ch 1, expanded dielectric table, 10232.eps, copied data from
Volume 1, Chapter 13; 10/2005.
• Nature of contribution: Mini table of contents, all chapters except appedicies; html,
latex, ps, pdf; See Devel/tutorial.html; 01/2006.
161.
A-2.2. CREDITS 153
• Nature of contribution: Changed CH2 from “Resistor color codes” to “Color codes”;
Added wiring color codes; 10/2007.
• Contact at: dcrunkilton(at)att(dot)net
A-2.2.2 Alejandro Gamero Divasto
• Date(s) of contribution(s): January 2002
• Nature of contribution: Suggestions related to troubleshooting: caveat regarding swap-
ping two similar components as a troubleshooting tool; avoiding pressure placed on the
troubleshooter; perils of ”team” troubleshooting; wisdom of recording system history; op-
erator error as a cause of failure; and the perils of ﬁnger-pointing.
A-2.2.3 Tony R. Kuphaldt
• Date(s) of contribution(s): 1996 to present
• Nature of contribution: Original author.
• Contact at: liec0@lycos.com
A-2.2.4 Your name here
• Date(s) of contribution(s): Month and year of contribution
• Nature of contribution: Insert text here, describing how you contributed to the book.
• Contact at: my email@provider.net
A-2.2.5 Typo corrections and other “minor” contributions
• The students of Bellingham Technical College’s Instrumentation program.
• Bernard Sheehan (January 2005), Typographical error correction in ”Right triangle
trigonometry” section Chapter 5: TRIGONOMETRY REFERENCE (two formulas for tan
x the second one reads tan x = cos x/sin x it should be cot x = cos x/sin x– changes to
01001.eps previously made)
• Michiel van Bolhuis (April 2007) Typo Ch 1, s/picofards/picofarads.
• Chirvasuta Constantin (April 2003) Identiﬁed error in quadratic equation formula.
• Colin Creitz (May 2007) Chapters: several, s/it’s/its.
• Jeff DeFreitas (March 2006)Improve appearance: replace “/” and ”/” Chapters: A1, A2.
• Gerald Gardner (January 2003) Suggested adding Imperial gallons conversion to table.
• Geoff Hosking (July 2006) Typo correction in Conductors and Insulators chapter, Criti-
cal Temperatures of Superconductors: s/degrees Kelvin/Kelvins.
162.
154 APPENDIX A-2. CONTRIBUTOR LIST
• Harvey Lew (??? 2003) Typo correction in Trig chapter: ”tangent” should have been
”cotangent”.
• Len Nunn (May 2008) Typo correction in Calculus chapter: ”dx/d(aˆx)” in error, 11042.png
.
• Don Stalkowski (June 2002) Technical help with PostScript-to-PDF ﬁle format conver-
sion.
• Joseph Teichman (June 2002) Suggestion and technical help regarding use of PNG
images instead of JPEG.
• Mark44@allaboutcircuits.com (March 2008) Ch 4, Clariﬁcation of division by zero.
• Timothy Unregistered@allaboutcircuits.com (Feb 2008) Changed default roman font
to newcent.
• Imranullah Syed (Feb 2008) Suggested centering of uncaptioned schematics.
• Unregistered@allaboutcircuits.com (Aug 2008) formatting of PDF off pps 130-136.
163.
Appendix A-3
DESIGN SCIENCE LICENSE
Copyright c 1999-2000 Michael Stutz stutz@dsl.org
Verbatim copying of this document is permitted, in any medium.
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Design Science License was written and deployed as a strategy for promoting the progress of
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155
164.
156 APPENDIX A-3. DESIGN SCIENCE LICENSE
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165.
A-3.5. 4. MODIFICATION 157
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166.
158 APPENDIX A-3. DESIGN SCIENCE LICENSE
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END OF TERMS AND CONDITIONS
[$Id: dsl.txt,v 1.25 2000/03/14 13:14:14 m Exp m $]
167.
Index
.end command, SPICE, 78 Derivative of a constant, 52
Electronics Workbench, 60 Derivative of power and log functions, 52
Derivative rules, 53
Addition method, simultaneous equations, 40 Dielectric strength, 27
Adjacent, 48 Difference, common, 34
Algebraic identities, 30 Differential Equations, 57
Ampacity, 24 Diodes, SPICE, 69
Analysis, AC, SPICE, 75
Analysis, DC, SPICE, 75 E, symbol for voltage, 2
Analysis, Fourier, SPICE, 76, 86
Analysis, transient, SPICE, 75 Factor, conversion, 12
Antiderivative of e functions, 56 Factorial, 35
Antiderivatives, 55 Factoring, 33
Arithmetic sequence, 34 Fault, ground, 122
FORTRAN, computer language, 61, 62
BASIC, computer language, 62
Gage size, wire, 23
C, computer language, 61 General solution, 57
Capacitance equation, 4 Geometric sequence, 35
Capacitors, SPICE, 68 Ground fault, 122
Common difference, 34
Common ratio, 35 Hyperbolic functions, 49
Compiler, 62 Hypotenuse, 48
Component names, SPICE, 67
Conductor ampacity, 24 I, symbol for current, 2
Constants, mathematical, 31 Impedance, 8
Conversion factor, 12 Independent variable, 57
Cosines, law of, 49 Inductance equation, 6
Critical temperature, high temperature super- Inductors, SPICE, 68
conductors, 26 Integral, deﬁnite, 56
Critical temperature, superconductors, 26 Integral, indeﬁnite, 55
Current measurement, SPICE, 83 Interpreter, 61
Current sources, AC, SPICE, 74
Current sources, DC, SPICE, 74 Joule’s Law, 2
Current sources, dependent, SPICE, 75
Current sources, pulse, SPICE, 74 Law of cosines, 49
Law of sines, 48
Derivative of e functions, 52 Limits, calculus, 52
159
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